Abstract

We examine the Goos-Hänchen (GH) effect for a Gaussian beam impinging on the surface of silicene whose topological phase transitions can be modulated by external electric field and/or irradiating circular polarized light. It is shown that both the spatial and angular shifts in GH effect present a sharp jump due to the topological phase transitions. The transitional GH effect can be attributed to transitional optical conductivity, which relates to Berry curvature and Chern numbers. These results can be extensively extended to other two-dimensional atomic crystals in graphene family. We believe that the transitional GH effect may offer a possible way to determine the Berry curvature, Chern numbers, and topological phase transition by a direct optical measurement.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  2. 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), 560–564 (2011).
    [Crossref] [PubMed]
  3. B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, “Epitaxial growth of a silicene sheet,” Appl. Phys. Lett. 97(22), 223109 (2010).
    [Crossref]
  4. C. C. Liu, W. X. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
    [Crossref] [PubMed]
  5. L. Chen, C. C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. G. Yao, and K. H. Wu, “Evidence for Dirac fermions in a honeycomb lattice based on silicon,” Phys. Rev. Lett. 109(5), 056804 (2012).
    [Crossref] [PubMed]
  6. P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. L. Lay, “Silicene: compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett. 108(15), 155501 (2012).
    [Crossref] [PubMed]
  7. S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two-and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
    [Crossref] [PubMed]
  8. C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
    [Crossref] [PubMed]
  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] [PubMed]
  10. M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012).
    [Crossref] [PubMed]
  11. M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
    [Crossref] [PubMed]
  12. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 436, 333–346 (1947).
    [Crossref]
  13. X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372 (2004).
    [Crossref]
  14. X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006).
    [Crossref]
  15. N. Hermosa, “Reflection beamshifts of visible light due to graphene,” J. Opt. 18(2), 025612 (2016).
    [Crossref]
  16. X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos-Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39(19), 5574–5577 (2014).
    [Crossref] [PubMed]
  17. X. Zeng, M. Al-Amri, and M. Suhail Zubairy, “Tunable Goos-Hänchen shift from graphene ribbon array,” Opt. Express 25(20), 23579 (2017).
    [Crossref] [PubMed]
  18. W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
    [Crossref]
  19. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
    [Crossref] [PubMed]
  20. X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012).
    [Crossref]
  21. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
    [Crossref]
  22. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86(1), 307–316 (2014).
    [Crossref]
  23. S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
    [Crossref]
  24. S. Grosche, A. Szameit, and M. Ornigotti, “Spatial Goos-Hänchen shift in photonic graphene,” Phys. Rev. A 94(6), 063831 (2016).
    [Crossref]
  25. M. Merano, A. Aiello, M. P. Van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
    [Crossref]
  26. H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96(24), 243903 (2006).
    [Crossref] [PubMed]
  27. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15(1), 014001 (2013).
    [Crossref]
  28. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33(13), 1437 (2008).
    [Crossref] [PubMed]
  29. D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, “Quantum spin-Hall effect and topologically invariant Chern numbers,” Phys. Rev. Lett. 97(3), 036808 (2006).
    [Crossref] [PubMed]
  30. E. Prodan, “Robustness of the spin-Chern number,” Phys. Rev. B 80(12), 125327 (2009).
    [Crossref]
  31. R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Jpn. 12(6), 570–586 (1957).
    [Crossref]
  32. L. Stille, C. J. Tabert, and E. J. Nicol, “Optical signatures of the tunable band gap and valley-spin coupling in silicene,” Phys. Rev. B 86(19), 195405 (2012).
    [Crossref]
  33. W. J. M. Kort-Kamp, “Topological phase transitions in the photonic spin Hall effect,” Phys. Rev. Lett. 119(14), 147401 (2017).
    [Crossref] [PubMed]
  34. P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, “Casimir force phase transitions in the graphene family,” Nat. Commun. 8, 14699 (2017)
    [Crossref] [PubMed]
  35. M. Merano, “Transverse electric surface mode in atomically thin boron-nitride,” Opt. Lett. 41(11), 2668–2671 (2016).
    [Crossref] [PubMed]
  36. M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
    [Crossref] [PubMed]
  37. M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
    [Crossref]
  38. L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
    [Crossref]
  39. W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
    [Crossref]
  40. F. D. M. Haldane, “Nobel lecture: Topological quantum matter,” Rev. Mod. Phys. 89(4), 040502 (2017).
    [Crossref]
  41. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
    [Crossref]
  42. B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
    [Crossref] [PubMed]

2018 (2)

W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
[Crossref]

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

2017 (8)

F. D. M. Haldane, “Nobel lecture: Topological quantum matter,” Rev. Mod. Phys. 89(4), 040502 (2017).
[Crossref]

X. Zeng, M. Al-Amri, and M. Suhail Zubairy, “Tunable Goos-Hänchen shift from graphene ribbon array,” Opt. Express 25(20), 23579 (2017).
[Crossref] [PubMed]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

W. J. M. Kort-Kamp, “Topological phase transitions in the photonic spin Hall effect,” Phys. Rev. Lett. 119(14), 147401 (2017).
[Crossref] [PubMed]

P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, “Casimir force phase transitions in the graphene family,” Nat. Commun. 8, 14699 (2017)
[Crossref] [PubMed]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

2016 (4)

S. Grosche, A. Szameit, and M. Ornigotti, “Spatial Goos-Hänchen shift in photonic graphene,” Phys. Rev. A 94(6), 063831 (2016).
[Crossref]

N. Hermosa, “Reflection beamshifts of visible light due to graphene,” J. Opt. 18(2), 025612 (2016).
[Crossref]

M. Merano, “Transverse electric surface mode in atomically thin boron-nitride,” Opt. Lett. 41(11), 2668–2671 (2016).
[Crossref] [PubMed]

M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
[Crossref] [PubMed]

2014 (3)

X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos-Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39(19), 5574–5577 (2014).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86(1), 307–316 (2014).
[Crossref]

2013 (2)

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15(1), 014001 (2013).
[Crossref]

M. Ezawa, “Photoinduced topological phase transition and a single Dirac-cone state in silicene,” Phys. Rev. Lett. 110(2), 026603 (2013).
[Crossref] [PubMed]

2012 (6)

M. Ezawa, “Valley-polarized metals and quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 109(5), 055502 (2012).
[Crossref] [PubMed]

L. Chen, C. C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. G. Yao, and K. H. Wu, “Evidence for Dirac fermions in a honeycomb lattice based on silicon,” Phys. Rev. Lett. 109(5), 056804 (2012).
[Crossref] [PubMed]

P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. L. Lay, “Silicene: compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett. 108(15), 155501 (2012).
[Crossref] [PubMed]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

L. Stille, C. J. Tabert, and E. J. Nicol, “Optical signatures of the tunable band gap and valley-spin coupling in silicene,” Phys. Rev. B 86(19), 195405 (2012).
[Crossref]

2011 (2)

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), 560–564 (2011).
[Crossref] [PubMed]

C. C. Liu, W. X. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
[Crossref] [PubMed]

2010 (1)

B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, “Epitaxial growth of a silicene sheet,” Appl. Phys. Lett. 97(22), 223109 (2010).
[Crossref]

2009 (3)

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two-and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

E. Prodan, “Robustness of the spin-Chern number,” Phys. Rev. B 80(12), 125327 (2009).
[Crossref]

M. Merano, A. Aiello, M. P. Van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

2008 (1)

2006 (4)

H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96(24), 243903 (2006).
[Crossref] [PubMed]

D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, “Quantum spin-Hall effect and topologically invariant Chern numbers,” Phys. Rev. Lett. 97(3), 036808 (2006).
[Crossref] [PubMed]

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), 1757–1761 (2006).
[Crossref] [PubMed]

X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006).
[Crossref]

2005 (2)

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

C. L. Kane and E. J. Mele, “Z2 Topological Order and the Quantum Spin Hall Effect,” Phys. Rev. Lett. 95(14), 146802 (2005).
[Crossref] [PubMed]

2004 (1)

X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372 (2004).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

1957 (1)

R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems,” J. Phys. Soc. Jpn. 12(6), 570–586 (1957).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 436, 333–346 (1947).
[Crossref]

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

Aiello, A.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15(1), 014001 (2013).
[Crossref]

M. Merano, A. Aiello, M. P. Van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33(13), 1437 (2008).
[Crossref] [PubMed]

Akturk, E.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two-and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Al-Amri, M.

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin −1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988).
[Crossref] [PubMed]

Asensio, M. C.

P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. L. Lay, “Silicene: compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett. 108(15), 155501 (2012).
[Crossref] [PubMed]

Aufray, B.

B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, “Epitaxial growth of a silicene sheet,” Appl. Phys. Lett. 97(22), 223109 (2010).
[Crossref]

Avila, J.

P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. L. Lay, “Silicene: compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett. 108(15), 155501 (2012).
[Crossref] [PubMed]

Bansil, A.

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), 560–564 (2011).
[Crossref] [PubMed]

Barr, L. E.

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

Bernevig, B. A.

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), 1757–1761 (2006).
[Crossref] [PubMed]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15(1), 014001 (2013).
[Crossref]

Boyd, R. W.

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86(1), 307–316 (2014).
[Crossref]

Cahangirov, S.

S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two-and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
[Crossref] [PubMed]

Cai, L.

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

Cava, R. J.

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), 560–564 (2011).
[Crossref] [PubMed]

Chen, J.

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

Chen, L.

L. Chen, C. C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. G. Yao, and K. H. Wu, “Evidence for Dirac fermions in a honeycomb lattice based on silicon,” Phys. Rev. Lett. 109(5), 056804 (2012).
[Crossref] [PubMed]

Chen, S.

W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
[Crossref]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

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L. Chen, C. C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. G. Yao, and K. H. Wu, “Evidence for Dirac fermions in a honeycomb lattice based on silicon,” Phys. Rev. Lett. 109(5), 056804 (2012).
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L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
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S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
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L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
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M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
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X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372 (2004).
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Lu, L.

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
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W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
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L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
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M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
[Crossref]

S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
[Crossref]

W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
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C. L. Kane and E. J. Mele, “Z2 Topological Order and the Quantum Spin Hall Effect,” Phys. Rev. Lett. 95(14), 146802 (2005).
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L. Chen, C. C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. G. Yao, and K. H. Wu, “Evidence for Dirac fermions in a honeycomb lattice based on silicon,” Phys. Rev. Lett. 109(5), 056804 (2012).
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S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
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W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
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J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Rev. Mod. Phys. 86(1), 307–316 (2014).
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B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, “Epitaxial growth of a silicene sheet,” Appl. Phys. Lett. 97(22), 223109 (2010).
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P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. L. Lay, “Silicene: compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett. 108(15), 155501 (2012).
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P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, “Casimir force phase transitions in the graphene family,” Nat. Commun. 8, 14699 (2017)
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S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin, and S. Ciraci, “Two-and one-dimensional honeycomb structures of silicon and germanium,” Phys. Rev. Lett. 102(23), 236804 (2009).
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H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96(24), 243903 (2006).
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D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, “Quantum spin-Hall effect and topologically invariant Chern numbers,” Phys. Rev. Lett. 97(3), 036808 (2006).
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L. Stille, C. J. Tabert, and E. J. Nicol, “Optical signatures of the tunable band gap and valley-spin coupling in silicene,” Phys. Rev. B 86(19), 195405 (2012).
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Wen, S.

W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
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W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
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X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
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X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012).
[Crossref]

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D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, “Quantum spin-Hall effect and topologically invariant Chern numbers,” Phys. Rev. Lett. 97(3), 036808 (2006).
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W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
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W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
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W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
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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), 1757–1761 (2006).
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W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
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W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
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X. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Large positive and negative lateral optical beam displacements due to surface plasmon resonance,” Appl. Phys. Lett. 85(3), 372 (2004).
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Zhou, X.

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
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X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012).
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X. Yin and L. Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89(26), 261108 (2006).
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X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
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B. Lalmi, H. Oughaddou, H. Enriquez, A. Kara, S. Vizzini, B. Ealet, and B. Aufray, “Epitaxial growth of a silicene sheet,” Appl. Phys. Lett. 97(22), 223109 (2010).
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S. Chen, C. Mi, L. Cai, M. Liu, H. Luo, and S. Wen, “Observation of the Goos-Hänchen shift in graphene via weak measurements,” Appl. Phys. Lett. 110(3), 031105 (2017).
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Opt. Express (1)

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W. Zhang, W. Wu, S. Chen, J. Zhang, X. Ling, W. Shu, H. luo, and S. Wen, “Photonic spin Hall effect on the surface of anisotropic two-dimensional atomic crystals,” Photonics Res. 6(6), 511 (2018).
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Phys. Rev. A (5)

M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spin-orbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A 95(6), 063827 (2017).
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L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
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X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012).
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W. Wu, S. Chen, C. Mi, W. Zhang, H. Luo, and S. Wen, “Giant quantized Goos-Hänchen effect on the surface of graphene in the quantum Hall regime,” Phys. Rev. A 96(4), 043814 (2017).
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L. Chen, C. C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. G. Yao, and K. H. Wu, “Evidence for Dirac fermions in a honeycomb lattice based on silicon,” Phys. Rev. Lett. 109(5), 056804 (2012).
[Crossref] [PubMed]

P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. L. Lay, “Silicene: compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett. 108(15), 155501 (2012).
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B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

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), 1757–1761 (2006).
[Crossref] [PubMed]

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), 560–564 (2011).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Schematic representation of the wave reflection at a freestanding silicene sheet. The freestanding silicene is subjected to a static electric field Ez and a circularly polarized lasers. The lattice constant and staggering length values are a = 3.86, and = 0.23, and the effective spin-orbit coupling λSO = 3.9meV. For the freestanding silicene, we can choose the refractive index of the substrate as n = 1.
Fig. 2
Fig. 2 Phase diagram of (a) longitudinal conductivity and (d) transverse Hall conductivity in the (Ez, Λ)/λSO plane, and the Chern number and spin-Chern number (��, ��s) are also indicated. The solid line and the dashed line together outline the phase boundaries indexed by Kη, where the solid line represents the spin up (s = +1) and the dashed line represents the spin down (s = −1). (b) Longitudinal conductivity, (e) transverse Hall conductivity, (c) horizontally polarized reflection coefficient, and (f) cross-polarized reflection coefficient is a function of the photon energy of circularly polarized light in the states of QSHI, PS-QHI, and QAHI. The parameters are Λ = 0λSO(blue), 1λSO(green), and 2λSO(red), Ezl = 0.5λSO, ħΓ = 0.002λSO, μ = 0.1λSO. And the solid (dashed) line represents the real (imaginary) part.
Fig. 3
Fig. 3 Transitional spatial GH effect in the (Ez, Λ)/λSO plane. The spatial shift, which is generated by (a) horizontally and (b) vertically polarized incident light, is a function of external electric field and coupling constants on the surface of silicene. The wavelength of incident light is λi = hc/E, incident angle θi = 60°, ħω = 0.1λSO, and the beam waist is chosen as w0 = 1mm. The (c) horizontally and (d) vertically polarized spatial shifts are modulated.
Fig. 4
Fig. 4 Transitional angular GH effect in the (Ez, Λ)/λSO plane. The angular deviation, which is generated by (a) horizontally and (b) vertically polarized incident light, is a function of external electric field and coupling constants on the surface of silicene. The (c) horizontally and (d) vertically polarized angular deviations are modulated. The parameters are the same as in Fig. 3.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

H η = ν F ( η k x τ x + k y τ y ) + λ S O σ z η τ z E z τ z Λ η τ z .
m s η = η s λ SO E z η Λ ,
σ ˜ x x η s σ 0 / 2 π = 4 μ 2 | m s η | 2 2 μ Ω Θ ( 2 μ | m s η | ) + ( 1 | m s η | 2 2 Ω 2 ) tan 1 ( Ω M ) + | m s η | 2 Ω M , σ ˜ x y η s σ 0 / 2 π = 2 η m s η Ω tan 1 ( Ω M ) .
r p p = α β + + σ p s σ s p α + β + + σ p s σ s p , r s s = α + β σ p s σ s p α + β + + σ p s σ s p , r p s = 2 Z i σ p s α + β + + σ p s σ s p , r s p = 2 Z i σ s p α + β + + σ p s σ s p .
E r exp ( i k i z r k i 2 x r 2 + y r 2 Z R + i z r ) × { x ^ r [ f p r p p ( 1 i x r Z R + i z r ln r p p θ i ) + f s r p s ( 1 i x r Z R + i z r ln r p s θ i ) ] + y ^ r [ f s r s s ( 1 i x r Z R + i z r ln r s s θ i ) + f p r s p ( 1 i x r Z R + i z r ln r p s θ i ) ] } ,
D GH = x r I ( x r , y r , z r ) d x r d y r I ( x r , y r , z r ) d x r d y r ,
D GH = R p p 2 φ p p + R s p 2 φ s p k i ( R s p 2 + R p p 2 ) z r R p p 2 ρ p p + R s p 2 ρ s p k i ( R s p 2 + R p p 2 ) Z R .
Δ GH P = R p p 2 φ p p + R s p 2 φ s p k i ( R p p 2 + R s p 2 ) ,
Δ GH SP = 1 k i φ s p .
Δ GH S = R s s 2 φ s s + R p s 2 φ p s k i ( R s s 2 + R p s 2 ) ,
Δ GH PS = 1 k i φ p s .
Θ GH P = 1 k i Z R R p p 2 ρ p p + R s p 2 ρ s p R s p 2 + R p p 2 .
Θ GH SP = 1 k i Z R ρ s p .
Θ GH S = 1 k i Z R R s s 2 ρ s s + R p s 2 ρ p s R p s 2 + R s s 2 ,
Θ GH PS = 1 k i Z R ρ p s .

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