Abstract

We show that a thin film of a three-dimensional topological insulator such as Bi2Se3 or Bi2Te3 can exhibit strong linear and nonlinear magneto-optical effects in a transverse magnetic field. In particular, one can achieve an almost complete circular polarization of an incident terahertz or mid-infrared radiation and an efficient four-wave mixing.

© 2015 Optical Society of America

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  1. M.Z. Hasan and C.L. Kane, “Colloquium: Topological insulators”, Mod. Phys. 82, 3045–3067 (2010).
    [Crossref]
  2. C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
    [Crossref]
  3. H.J. 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, 438–442 (2009).
    [Crossref]
  4. 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. 5398–402 (2009).
    [Crossref]
  5. X. Zhang, J. Wang, and S.-C. Zhang, “Topological insulators for high-performance terahertz to infrared applications,” Phys. Rev. B 82, 245107 (2010).
    [Crossref]
  6. T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
    [Crossref]
  7. A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
    [Crossref]
  8. P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
    [Crossref]
  9. Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
    [Crossref]
  10. R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
    [Crossref]
  11. D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
    [Crossref]
  12. W.-K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr Effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
    [Crossref] [PubMed]
  13. J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
    [Crossref]
  14. F. Liu, J. Xu, and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 41, 735–741 (2014).
    [Crossref]
  15. W.-Y. Shan, H. Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators,” New J. of Phys. 12, 043048 (2010).
    [Crossref]
  16. H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
    [Crossref]
  17. Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
    [Crossref]
  18. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2, 1979).
  19. X. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012).
    [Crossref] [PubMed]
  20. M. Tokman, X. Yao, and A. Belyanin, “Generation of entangled states in graphene in a strong magnetic field,” Phys. Rev. Lett. 110, 0774904 (2013).
    [Crossref]
  21. X. Yao and A. Belyanin, “Nonlinear optics of graphene in a strong magnetic field,” J. Phys. Cond. Matt. 25, 054203 (2013).
    [Crossref]

2014 (2)

F. Liu, J. Xu, and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 41, 735–741 (2014).
[Crossref]

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

2013 (2)

M. Tokman, X. Yao, and A. Belyanin, “Generation of entangled states in graphene in a strong magnetic field,” Phys. Rev. Lett. 110, 0774904 (2013).
[Crossref]

X. Yao and A. Belyanin, “Nonlinear optics of graphene in a strong magnetic field,” J. Phys. Cond. Matt. 25, 054203 (2013).
[Crossref]

2012 (4)

X. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012).
[Crossref] [PubMed]

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

2010 (9)

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

X. Zhang, J. Wang, and S.-C. Zhang, “Topological insulators for high-performance terahertz to infrared applications,” Phys. Rev. B 82, 245107 (2010).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

M.Z. Hasan and C.L. Kane, “Colloquium: Topological insulators”, Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

W.-K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr Effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref] [PubMed]

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

W.-Y. Shan, H. Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators,” New J. of Phys. 12, 043048 (2010).
[Crossref]

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

2009 (3)

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

H.J. 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, 438–442 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Aguilar, R. V.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Ando, Y.

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Armitage, N. P.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Bansal, N.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Bansil, A.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Basov, D. N.

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Belyanin, A.

M. Tokman, X. Yao, and A. Belyanin, “Generation of entangled states in graphene in a strong magnetic field,” Phys. Rev. Lett. 110, 0774904 (2013).
[Crossref]

X. Yao and A. Belyanin, “Nonlinear optics of graphene in a strong magnetic field,” J. Phys. Cond. Matt. 25, 054203 (2013).
[Crossref]

X. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012).
[Crossref] [PubMed]

Bilbro, L. S.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Cava, R. J.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Cerne, J.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Chapler, B. C.

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Chen, M.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

Chen, X.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

Chen, Xi

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Cheng, P.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Dai, X.

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

H.J. 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, 438–442 (2009).
[Crossref]

Dai, Xi

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Dil, J. H.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Drew, H. D.

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

Fang, Z.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

H.J. 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, 438–442 (2009).
[Crossref]

Fedorov, A. V.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Fu, Y.-S.

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

George, D. K.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Grauer, D.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Hanaguri, T.

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

Hasan, M. Z.

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. 5398–402 (2009).
[Crossref]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Hasan, M.Z.

M.Z. Hasan and C.L. Kane, “Colloquium: Topological insulators”, Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

He, K.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

He, Ke

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Hor, Y. S.

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. 5398–402 (2009).
[Crossref]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Hsieh, D.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Igarashi, K.

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

Jia, Jin-Feng

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Jiang, Y.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

Kane, C.L.

M.Z. Hasan and C.L. Kane, “Colloquium: Topological insulators”, Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Kawamura, M.

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2, 1979).

Li, Z.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2, 1979).

Lin, H.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Liu, C.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Liu, C.X.

H.J. 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, 438–442 (2009).
[Crossref]

Liu, C.-X.

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

Liu, F.

F. Liu, J. Xu, and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 41, 735–741 (2014).
[Crossref]

Liu, W.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Lu, H.

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

Lu, H. Z.

W.-Y. Shan, H. Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators,” New J. of Phys. 12, 043048 (2010).
[Crossref]

Ma, X.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

MacDonald, A. H.

W.-K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr Effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref] [PubMed]

Maciejko, J.

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

Markelz, A. G.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Meier, F.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Niu, Q.

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

Oh, S.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Osterwalder, J.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Pal, A.

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. 5398–402 (2009).
[Crossref]

Patthey, L.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Pitaevski, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2, 1979).

Post, K. W.

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Qi, X.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Qi, X.L.

H.J. 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, 438–442 (2009).
[Crossref]

Qi, X.-L.

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

Qi, Xiao-Liang

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Qian, D.

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. 5398–402 (2009).
[Crossref]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Sasagawa, T.

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

Schafgans, A. A.

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Shan, W.

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

Shan, W.-Y.

W.-Y. Shan, H. Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators,” New J. of Phys. 12, 043048 (2010).
[Crossref]

Shen, S.-Q.

W.-Y. Shan, H. Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators,” New J. of Phys. 12, 043048 (2010).
[Crossref]

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

Song, C.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Stier, A.V.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Takagi, H.

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

Taskin, A. A.

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Tokman, M.

M. Tokman, X. Yao, and A. Belyanin, “Generation of entangled states in graphene in a strong magnetic field,” Phys. Rev. Lett. 110, 0774904 (2013).
[Crossref]

Tse, W.-K.

W.-K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr Effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref] [PubMed]

Wang, J.

X. Zhang, J. Wang, and S.-C. Zhang, “Topological insulators for high-performance terahertz to infrared applications,” Phys. Rev. B 82, 245107 (2010).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Wang, L.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Wang, Y.

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Wray, L.

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. 5398–402 (2009).
[Crossref]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

Wu, L.

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

Xia, Y.

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Xie, X.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Xu, J.

F. Liu, J. Xu, and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 41, 735–741 (2014).
[Crossref]

Xue, Q.-K.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Xue, Qi-Kun

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

Yang, Y.

F. Liu, J. Xu, and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 41, 735–741 (2014).
[Crossref]

Yao, W.

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

Yao, X.

M. Tokman, X. Yao, and A. Belyanin, “Generation of entangled states in graphene in a strong magnetic field,” Phys. Rev. Lett. 110, 0774904 (2013).
[Crossref]

X. Yao and A. Belyanin, “Nonlinear optics of graphene in a strong magnetic field,” J. Phys. Cond. Matt. 25, 054203 (2013).
[Crossref]

X. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012).
[Crossref] [PubMed]

Zhang, H.

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

Zhang, H.J.

H.J. 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, 438–442 (2009).
[Crossref]

Zhang, S.-C.

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

X. Zhang, J. Wang, and S.-C. Zhang, “Topological insulators for high-performance terahertz to infrared applications,” Phys. Rev. B 82, 245107 (2010).
[Crossref]

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

H.J. 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, 438–442 (2009).
[Crossref]

Zhang, T.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Zhang, X.

X. Zhang, J. Wang, and S.-C. Zhang, “Topological insulators for high-performance terahertz to infrared applications,” Phys. Rev. B 82, 245107 (2010).
[Crossref]

Zhang, Y.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Zhu, B. F.

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

J. Opt. Soc. Am. B (1)

F. Liu, J. Xu, and Y. Yang, “Polarization conversion of reflected electromagnetic wave from topological insulator,” J. Opt. Soc. Am. B 41, 735–741 (2014).
[Crossref]

J. Phys. Cond. Matt. (1)

X. Yao and A. Belyanin, “Nonlinear optics of graphene in a strong magnetic field,” J. Phys. Cond. Matt. 25, 054203 (2013).
[Crossref]

Mod. Phys. (1)

M.Z. Hasan and C.L. Kane, “Colloquium: Topological insulators”, Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Nat. Phys. (2)

H.J. 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, 438–442 (2009).
[Crossref]

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. 5398–402 (2009).
[Crossref]

Nature Phys. (1)

Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, “Imaging the two-component nature of DiracLandau levels in the topological surface state of Bi2Se3,” Nature Phys. 10, 815–819 (2014).
[Crossref]

New J. of Phys. (1)

W.-Y. Shan, H. Z. Lu, and S.-Q. Shen, “Effective continuous model for surface states and thin films of three-dimensional topological insulators,” New J. of Phys. 12, 043048 (2010).
[Crossref]

Phys. Rev. B (5)

H. Lu, W. Shan, W. Yao, Q. Niu, and S.-Q. Shen, “Massive Dirac fermions and spin physics in an ultrathin film of topological insulator,” Phys. Rev. B 81, 115407 (2010).
[Crossref]

C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, “Model Hamiltonian for topological insulators,” Phys. Rev. B 82, 045122 (2010).
[Crossref]

X. Zhang, J. Wang, and S.-C. Zhang, “Topological insulators for high-performance terahertz to infrared applications,” Phys. Rev. B 82, 245107 (2010).
[Crossref]

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, “Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3,” Phys. Rev. B 82, 081305 (2010).
[Crossref]

A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, Xiao-Liang Qi, B. C. Chapler, and D. N. Basov, “Landau level spectroscopy of surface states in the topological insulator Bi0.91Sb0.09 via magneto-optics,” Phys. Rev. B 85, 195440 (2012).
[Crossref]

Phys. Rev. Lett. (8)

P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, Jin-Feng Jia, J. Wang, Y. Wang, B. F. Zhu, Xi Chen, X. Ma, Ke He, L. Wang, Xi Dai, Z. Fang, X. Xie, X. Qi, C. Liu, S.-C. Zhang, and Q.-K. Xue, “Landau quantization of topological surface states in Bi2Se3,” Phys. Rev. Lett. 105, 076801 (2010).
[Crossref]

Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Qi-Kun Xue, “Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3,” Phys. Rev. Lett. 108, 016401 (2012).
[Crossref]

R. V. Aguilar, A.V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, “Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3,” Phys. Rev. Lett. 108, 087403 (2012).
[Crossref]

D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of time-reversal-protected single- Dirac-cone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103, 146401 (2009).
[Crossref]

W.-K. Tse and A. H. MacDonald, “Giant magneto-optical Kerr Effect and universal Faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref] [PubMed]

J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

X. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” Phys. Rev. Lett. 108, 255503 (2012).
[Crossref] [PubMed]

M. Tokman, X. Yao, and A. Belyanin, “Generation of entangled states in graphene in a strong magnetic field,” Phys. Rev. Lett. 110, 0774904 (2013).
[Crossref]

Other (1)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2, 1979).

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Figures (6)

Fig. 1
Fig. 1 (a) Energy bands of the surface states in Bi2Se3 near the Γ point at zero magnetic field (with quadratic correction shown by dashed line), and energies of Landau levels in a magnetic field of 10 T (horizontal lines). (b) Landau level energies as a function of the magnetic field for Bi2Se3, for transitions between states n = 1 and 2 (red bottom line), 0 and 1 (black middle line), and -1 and 2 (blue top line).
Fig. 2
Fig. 2 (a) An example of the experimental geometry: the incident field is linearly polarized with orientation angle π 4. (b)The optical transition scheme for incident frequency ω ≈ ωc. Here Fermi level is placed between Landau levels -1 and 0. (c) Polarization coefficient K of the transmitted optical field, as a function of ξ = ω c 4 γ e 2 ħ c. The slab thickness is chosen as 0.01 λ in the plot.
Fig. 3
Fig. 3 (a) Multiple reflections in the slab geometry. (b) The real and imaginary part of the polarization coefficient (K) of the transmitted optical field as a function of the slab thickness d. (c) Transmittance T as a function of slab thickness d (solid line). The dashed line is for a pure dielectric slab without surface layers of massless fermions. Here ξ = 4 in both plots.
Fig. 4
Fig. 4 Polarization ellipse of the transmitted field. First row: from left to right, ξ = 0.1,1,2,3,4; d is fixed at 0.01λ. Second row: from left to right, d = 0.01λ,0.2λ,0.29λ,0.4λ,0.5λ; ξ is fixed at 4. The sense of rotation in the polarization is indicated by arrows.
Fig. 5
Fig. 5 Landau levels near the Dirac point superimposed on the electron dispersion without the magnetic field E = ±υF|p|. (b): A scheme of the four-wave mixing process in the four-level system of Landau levels with energy quantum numbers n = 1,0,+1,+2 that are renamed to states 1 through 4 for convenience.
Fig. 6
Fig. 6 Top panel: Intensity of the four wave mixing signal E3 as a function of the normalized intensity of the pump field E1, x = I1/Isat where the saturation intensity Isat ≈ 104 W/cm2 in the magnetic field of 1T and for a scattering rate γ = 1012 s1. The normalized intensity of the second pump field E2 is taken as 0.6x. Bottom panel: the Raman gain G for the field E3 under the same conditions and in the absence of the second field E2.

Equations (27)

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H ( k ) = ( C D 1 z 2 + D 2 k 2 ) + ( h ( A 1 ) A 2 k σ x A 2 k + σ x h ( A 1 ) )
h ( A 1 ) = ( M + B 1 z 2 B 2 k 2 ) σ z i A 1 z σ x ,
H ( k ) = ( h + ( k ) 0 0 h ( k ) ) ;
h ± ( k ) = E 0 D k 2 + ħ v F ( σ × k ) z ± σ z ( Δ / 2 B k 2 ) ,
H e f f ( k ) = E 0 D k 2 + ħ v F ( k y σ x k x σ y ) ,
H e f f ( π ) = E 0 2 D l c 2 ( a a + 1 2 ) + 2 ħ v F l c ( 0 i a i a 0 ) .
Ψ n = C n L e i k y y ( sgn ( n ) · i | n | ψ | n | 1 , i | n | 1 ψ | n | ) T , E n = E 0 2 | n | D l c 2 + sgn ( n ) 2 | n | ( ħ v F l c ) 2 + ( D l c ) 2 .
H e f f ( π ) = E 0 ħ v F ( π x σ y π y σ x ) .
H i n t = v F ( σ × A o p t ) z ( μ ~ × E ( ω ) ) z e i ω t ,
μ m n = e m | r | n = i e ħ v f F E n E m m | σ x y ^ σ y x ^ | n .
P ( r , t ) = N · tr ( ρ ^ · μ ) .
ρ ˙ n m = i ħ ( ε n ε m ) ρ n m i ħ [ H ^ i n t ( t ) , ρ ^ ] n m γ n m ( ρ n m ρ n m ( e q ) ) .
P ( 1 ) ( ω ) = N n m ρ m n ( e q ) ρ n n ( e q ) ħ · ( μ ~ n m × e ^ ) z μ m n ( ω n m ω ) i γ n m · E ( ω ) exp ( i ω t ) .
χ k j ( 1 ) ( ω ) = N ρ m m ( e q ) ρ n n ( e q ) 2 ħ ε z i j μ ˜ n m i μ m n k ω n m ω i γ ,
χ ˜ = ( χ x x χ x y χ y x χ y y ) = ( χ i g i g χ ) ,
χ = Ω n m 2 ω n m 2 ( ω + i γ ) 2 ; g = s · ω + i γ ω n m Ω n m 2 ω n m 2 ( ω + i γ ) 2 ,
Ω n m 2 = C m 2 C n 2 ( ρ m m ( e q ) ρ n n ( e q ) ) N e 2 v F 2 ħ ω , s = { + 1 | n | = | m | 1 1 | m | = | n | 1 ; C n = { 1 ( n = 0 ) 1 2 ( n 0 ) .
( E y E x ) = ( 1 α χ x x α χ y x α χ x y 1 α χ x y ) ( 1 α χ x x ) 2 + ( α χ x y ) 2 ( E y E x ) ,
K = K ( 1 α χ x x ) + α χ y x K α χ x y + ( 1 α χ y y ) .
T 1 = 2 ( n + 1 2 α χ 2 α i g 2 α i g n + 1 2 α χ ) ( n + 1 2 α χ ) 2 ( 2 α g ) 2
T 2 = 2 ( n 2 + n 2 n α χ 2 n α i g 2 n α i g n 2 + n 2 n α χ ) ( 1 + n 2 α χ ) 2 ( 2 α g ) 2
( E y E x ) = e i n ω L c T 2 ( I e 2 i n ω L c R 2 R 2 ) 1 T 1 ( E y E x )
μ = e l c 2 ( 0 i x ^ y ^ 0 1 x ^ y ^ 2 + 2 i x ^ y ^ 0 i x ^ y ^ 0 0 i x ^ y ^ 0 1 x ^ y ^ 2 2 1 x ^ y ^ 2 + 2 0 1 x ^ y ^ 2 2 0 )
χ ( 3 ) ( ω 3 ) = N μ 43 μ ˜ 41 μ ˜ 32 * μ ˜ 21 * ( i ħ ) 3 Γ 43 ( ρ 22 ρ 33 Γ 31 * Γ 32 * + ρ 22 ρ 11 Γ 31 * Γ 21 * ρ 11 ρ 44 Γ 42 Γ 41 + ρ 22 ρ 11 Γ 42 Γ 21 * ) .
E z = i · 2 π ω c · P .
E 3 = 2 π i ω 3 c χ ( 3 ) E 1 ( E 2 * ) 2 .
G = 2 π ω 3 N μ 43 μ ˜ 43 ħ c Γ 43 ( n 43 | Ω 41 | 2 Γ 31 * Γ 41 * n 41 ) × 1 / ( 1 + | Ω 41 | 2 / ( Γ 43 Γ 31 * ) ) ,

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