Abstract

We investigate the propagation of surface plasmon polaritons (SPPs) in thin films of topological insulators. Cases of single films and multilayered stacks are analyzed. The materials considered are second generation three dimensional topological insulators Bi2Se3, Bi2Te3, and Sb2Te3. Dispersion relations and propagation lengths of SPPs are estimated numerically, taking into account the variation of bulk dielectric functions of topological insulators, as well as substrate, using the Drude-Lorentz model. The key factors affecting propagation length are identified and experimental modifications for tuning the dispersion relations are proposed. The apparent discrepancy between the experimental data and previously considered theory is resolved.

© 2016 Optical Society of America

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    [Crossref]

2015 (5)

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
[Crossref] [PubMed]

Z. Chen, L. Zhao, K. Park, T. A. Garcia, M. C. Tamargo, and L. Krusin-Elbaum, “Robust topological interfaces and charge transfer in epitaxial Bi2Se3/II–VI semiconductor superlattices,” Nano Lett. 15(10), 6365–6370 (2015).
[Crossref] [PubMed]

2014 (6)

D. A. Smirnova, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Multilayer graphene waveguides,” JETP Lett. 99(8), 456–460 (2014).
[Crossref]

Y. P. Lai, I.T. Lin, K. Wu, and J. M. Liu, “Plasmonics in topological insulators,” Nanomater. and Nanotecho. 4(13), 1–8 (2014).
[Crossref]

S. Sim, M. Brahlek, N. Koirala, S. Cha, S. Oh, and H. Choi, “Ultrafast terahertz dynamics of hot Dirac-electron surface scattering in the topological insulator Bi2Se3,” Phys. Rev. B 89, 165137 (2014).
[Crossref]

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

D. Kim, S. Paul, N. P. Butch, J. Paglione, and M. S. Fuhrer, “Ambipolar surface state thermoelectric power of topological insulator Bi2Se3,” Nano Lett. 141701–1706 (2014).
[Crossref]

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

2013 (3)

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

T. Stauber, G. Gomez-Santós, and L. Brey, “Spin-charge separation of plasmonic excitations in thin topological insulators,” Phys. Rev. B 88(20), 205427 (2013).
[Crossref]

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jap. 82(10), 102001 (2013).
[Crossref]

2012 (4)

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (2012).
[Crossref]

V. R. 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(8), 087403 (2012).
[Crossref]

R. E. V. Profumo, R. Asgari, M. Polini, and A. H. MacDonald, “Double-layer graphene and topological insulator thin-film plasmons,” Phys. Rev. B 85(8), 085443 (2012).
[Crossref]

2011 (1)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

2010 (3)

M. Z. Hasan and C. L. Kane, “Colloquium : topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

S. Raghu, S. B. Chung, X. L. Qi, and S. C. Zhang, “Collective modes of a helical liquid,” Phys. Rev. Lett. 104(11), 116401 (2010)
[Crossref] [PubMed]

O. V. Yazyev, J. E. Moore, and S. G. Louie, “Spin polarization and transport of surface states in the topological insulators Bi2Se3 and Bi2Te3 from first principles,” Phys. Rev. Lett. 105(26), 266806 (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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

S. Das Sarma and E. H. Hwang, “Collective modes of the massless Dirac plasma,” Phys. Rev. Lett. 102(20), 206412 (2009).
[Crossref] [PubMed]

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, 438–442 (2009).
[Crossref]

2007 (1)

L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions,” Phys. Rev. Lett. 98(10), 106803 (2007).
[Crossref] [PubMed]

2001 (1)

A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp. 44(10S), 131 (2001).
[Crossref]

2000 (1)

M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61(12), 8187–8201 (2000).
[Crossref]

1999 (1)

El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
[Crossref]

1991 (1)

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

1977 (1)

W. Richter, H. Kohler, and C. R. Becker, “A Raman and far-infrared investigation of phonons in the rhombohedral V2-VI3 compounds,” Phys. Stat. Sol. (b) 84, 619–628 (1977).
[Crossref]

1962 (1)

Adamo, G.

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

Aguilar, V. R.

V. R. 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(8), 087403 (2012).
[Crossref]

Analytis, J. G.

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), 178–181 (2009).
[PubMed]

Ando, Y.

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jap. 82(10), 102001 (2013).
[Crossref]

Armitage, N. P.

V. R. 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(8), 087403 (2012).
[Crossref]

Asgari, R.

R. E. V. Profumo, R. Asgari, M. Polini, and A. H. MacDonald, “Double-layer graphene and topological insulator thin-film plasmons,” Phys. Rev. B 85(8), 085443 (2012).
[Crossref]

Autore, M.

M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

Awschalom, D. D.

A. L. Yeats, Y. Pan, A. Richardella, P. J. Mintun, N. Samarth, and D. D. Awschalom, “Persistent optical gating of a topological insulator,” Sci. Adv.1 (2014).

Baldassarre, L.

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (2012).
[Crossref]

Bansal, N.

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

V. R. 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(8), 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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Becker, C. R.

W. Richter, H. Kohler, and C. R. Becker, “A Raman and far-infrared investigation of phonons in the rhombohedral V2-VI3 compounds,” Phys. Stat. Sol. (b) 84, 619–628 (1977).
[Crossref]

Benet, S.

El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
[Crossref]

Bilbro, L. S.

V. R. 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(8), 087403 (2012).
[Crossref]

Bosman, M.

M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
[Crossref] [PubMed]

Brahlek, M.

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

S. Sim, M. Brahlek, N. Koirala, S. Cha, S. Oh, and H. Choi, “Ultrafast terahertz dynamics of hot Dirac-electron surface scattering in the topological insulator Bi2Se3,” Phys. Rev. B 89, 165137 (2014).
[Crossref]

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

Brey, L.

T. Stauber, G. Gomez-Santós, and L. Brey, “Spin-charge separation of plasmonic excitations in thin topological insulators,” Phys. Rev. B 88(20), 205427 (2013).
[Crossref]

Butch, N. P.

D. Kim, S. Paul, N. P. Butch, J. Paglione, and M. S. Fuhrer, “Ambipolar surface state thermoelectric power of topological insulator Bi2Se3,” Nano Lett. 141701–1706 (2014).
[Crossref]

Calvani, P.

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (2012).
[Crossref]

Cava, R.

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

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V. R. 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(8), 087403 (2012).
[Crossref]

Cha, S.

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

S. Sim, M. Brahlek, N. Koirala, S. Cha, S. Oh, and H. Choi, “Ultrafast terahertz dynamics of hot Dirac-electron surface scattering in the topological insulator Bi2Se3,” Phys. Rev. B 89, 165137 (2014).
[Crossref]

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El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
[Crossref]

Chen, C.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Chen, Y. L.

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), 178–181 (2009).
[PubMed]

Chen, Z.

Z. Chen, L. Zhao, K. Park, T. A. Garcia, M. C. Tamargo, and L. Krusin-Elbaum, “Robust topological interfaces and charge transfer in epitaxial Bi2Se3/II–VI semiconductor superlattices,” Nano Lett. 15(10), 6365–6370 (2015).
[Crossref] [PubMed]

Choi, H.

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

S. Sim, M. Brahlek, N. Koirala, S. Cha, S. Oh, and H. Choi, “Ultrafast terahertz dynamics of hot Dirac-electron surface scattering in the topological insulator Bi2Se3,” Phys. Rev. B 89, 165137 (2014).
[Crossref]

Chu, J.-H.

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), 178–181 (2009).
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S. Raghu, S. B. Chung, X. L. Qi, and S. C. Zhang, “Collective modes of a helical liquid,” Phys. Rev. Lett. 104(11), 116401 (2010)
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M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

Dai, X.

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, 438–442 (2009).
[Crossref]

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), 178–181 (2009).
[PubMed]

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M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
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M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

Deneuville, A.

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
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Di Gaspare, A.

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

Di Pietro, P.

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (2012).
[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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

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Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Engelkamp, H.

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

Fang, Z.

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, 438–442 (2009).
[Crossref]

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), 178–181 (2009).
[PubMed]

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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Feng, Y.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Fisher, I. R.

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), 178–181 (2009).
[PubMed]

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L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions,” Phys. Rev. Lett. 98(10), 106803 (2007).
[Crossref] [PubMed]

Fuhrer, M. S.

D. Kim, S. Paul, N. P. Butch, J. Paglione, and M. S. Fuhrer, “Ambipolar surface state thermoelectric power of topological insulator Bi2Se3,” Nano Lett. 141701–1706 (2014).
[Crossref]

Garcia, T. A.

Z. Chen, L. Zhao, K. Park, T. A. Garcia, M. C. Tamargo, and L. Krusin-Elbaum, “Robust topological interfaces and charge transfer in epitaxial Bi2Se3/II–VI semiconductor superlattices,” Nano Lett. 15(10), 6365–6370 (2015).
[Crossref] [PubMed]

Gardner, D. R.

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

Gedik, N.

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

George, D. K.

V. R. 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(8), 087403 (2012).
[Crossref]

Giliberti, V.

M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

Giorgianni, F.

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

Gomez-Santós, G.

T. Stauber, G. Gomez-Santós, and L. Brey, “Spin-charge separation of plasmonic excitations in thin topological insulators,” Phys. Rev. B 88(20), 205427 (2013).
[Crossref]

Gratens, X.

El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
[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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium : topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
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Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
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Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
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M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61(12), 8187–8201 (2000).
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Holloway, P. H.

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

Hor, Y. S.

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Hsieh, D.

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
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Hussain, Z.

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), 178–181 (2009).
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S. Das Sarma and E. H. Hwang, “Collective modes of the massless Dirac plasma,” Phys. Rev. Lett. 102(20), 206412 (2009).
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D. A. Smirnova, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Multilayer graphene waveguides,” JETP Lett. 99(8), 456–460 (2014).
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Jarillo-Herrero, P.

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

Jo, M.-H.

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

Kaddouri, El. H.

El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium : topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions,” Phys. Rev. Lett. 98(10), 106803 (2007).
[Crossref] [PubMed]

Kim, D.

D. Kim, S. Paul, N. P. Butch, J. Paglione, and M. S. Fuhrer, “Ambipolar surface state thermoelectric power of topological insulator Bi2Se3,” Nano Lett. 141701–1706 (2014).
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D. A. Smirnova, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Multilayer graphene waveguides,” JETP Lett. 99(8), 456–460 (2014).
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Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

Sim, S.

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

S. Sim, M. Brahlek, N. Koirala, S. Cha, S. Oh, and H. Choi, “Ultrafast terahertz dynamics of hot Dirac-electron surface scattering in the topological insulator Bi2Se3,” Phys. Rev. B 89, 165137 (2014).
[Crossref]

Smirnova, D. A.

D. A. Smirnova, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Multilayer graphene waveguides,” JETP Lett. 99(8), 456–460 (2014).
[Crossref]

So, J.-K.

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

Song, P.

M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
[Crossref] [PubMed]

Stauber, T.

T. Stauber, G. Gomez-Santós, and L. Brey, “Spin-charge separation of plasmonic excitations in thin topological insulators,” Phys. Rev. B 88(20), 205427 (2013).
[Crossref]

Steinberg, H.

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

Stier, A. V.

V. R. 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(8), 087403 (2012).
[Crossref]

Sulaev, A.

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

Sung, J. H.

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

Tamargo, M. C.

Z. Chen, L. Zhao, K. Park, T. A. Garcia, M. C. Tamargo, and L. Krusin-Elbaum, “Robust topological interfaces and charge transfer in epitaxial Bi2Se3/II–VI semiconductor superlattices,” Nano Lett. 15(10), 6365–6370 (2015).
[Crossref] [PubMed]

Tanner, D.

A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

Tedenac, J. C.

El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
[Crossref]

Tiwald, T. E.

M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61(12), 8187–8201 (2000).
[Crossref]

Vecchio, I. L.

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
[Crossref]

Vitucci, F. M.

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (2012).
[Crossref]

Wang, L.

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

Wang, X.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Wang, Y. H.

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

Wang, Z.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Wolf, M.S.

M.S. Wolf, Infrared and Optical Studies of Topological Insulators Bi2Se3, Bi2Te3 and Sb2Te3, (Thesis, The University of Akron, 2011).

Wray, 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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Wu, K.

Y. P. Lai, I.T. Lin, K. Wu, and J. M. Liu, “Plasmonics in topological insulators,” Nanomater. and Nanotecho. 4(13), 1–8 (2014).
[Crossref]

Wu, L.

V. R. 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(8), 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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Xie, Z.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Xu, Z.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Yang, F.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Yazyev, O. V.

O. V. Yazyev, J. E. Moore, and S. G. Louie, “Spin polarization and transport of surface states in the topological insulators Bi2Se3 and Bi2Te3 from first principles,” Phys. Rev. Lett. 105(26), 266806 (2010).
[Crossref]

Yeats, A. L.

A. L. Yeats, Y. Pan, A. Richardella, P. J. Mintun, N. Samarth, and D. D. Awschalom, “Persistent optical gating of a topological insulator,” Sci. Adv.1 (2014).

Yi, H.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Yu, L.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Zeng, M.

M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
[Crossref] [PubMed]

Zhang, F.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Zhang, H.

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, 438–442 (2009).
[Crossref]

Zhang, H. J.

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), 178–181 (2009).
[PubMed]

Zhang, J.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Zhang, S.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Zhang, S. C.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

S. Raghu, S. B. Chung, X. L. Qi, and S. C. Zhang, “Collective modes of a helical liquid,” Phys. Rev. Lett. 104(11), 116401 (2010)
[Crossref] [PubMed]

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, 438–442 (2009).
[Crossref]

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), 178–181 (2009).
[PubMed]

Zhao, L.

Z. Chen, L. Zhao, K. Park, T. A. Garcia, M. C. Tamargo, and L. Krusin-Elbaum, “Robust topological interfaces and charge transfer in epitaxial Bi2Se3/II–VI semiconductor superlattices,” Nano Lett. 15(10), 6365–6370 (2015).
[Crossref] [PubMed]

Zhao, M.

M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
[Crossref] [PubMed]

Zheludev, N. I.

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

Zhou, X. J.

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Adv. Opt. Mat. (1)

M. Autore, F. D’Apuzzo, A. Di Gaspare, V. Giliberti, O. Limaj, P. Roy, M. Brahlek, N. Koirala, S. Oh, F. J. G. de Abajo, and S. Lupi, “Plasmon-phonon interactions in topological insulator microrings,” Adv. Opt. Mat. 3, 1257–1263 (2015).
[Crossref]

ASC Photonics (1)

M. Autore, H. Engelkamp, F. D’Apuzzo, A. Di Gaspare, P. Di Pietro, I. L. Vecchio, M. Brahlek, N. Koirala, S. Oh, and S. Lupi, “Observation of magnetoplasmons in Bi2Se3 topological insulator,” ASC Photonics 2, 1231–1235 (2015).
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J. Opt. Soc. Am. (1)

J. Phys. Soc. Jap. (1)

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jap. 82(10), 102001 (2013).
[Crossref]

JETP Lett. (1)

D. A. Smirnova, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, “Multilayer graphene waveguides,” JETP Lett. 99(8), 456–460 (2014).
[Crossref]

Nano Lett. (3)

Z. Chen, L. Zhao, K. Park, T. A. Garcia, M. C. Tamargo, and L. Krusin-Elbaum, “Robust topological interfaces and charge transfer in epitaxial Bi2Se3/II–VI semiconductor superlattices,” Nano Lett. 15(10), 6365–6370 (2015).
[Crossref] [PubMed]

D. Kim, S. Paul, N. P. Butch, J. Paglione, and M. S. Fuhrer, “Ambipolar surface state thermoelectric power of topological insulator Bi2Se3,” Nano Lett. 141701–1706 (2014).
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M. Zhao, M. Bosman, M. Danesh, M. Zeng, P. Song, Y. Darma, A. Rusydi, H. Lin, C.-W. Qiu, and K. P. Loh, “Visible surface plasmon modes in single Bi2Te3 nanoplate,” Nano Lett. 15(12), 8331–8335 (2015).
[Crossref] [PubMed]

Nanomater. and Nanotecho. (1)

Y. P. Lai, I.T. Lin, K. Wu, and J. M. Liu, “Plasmonics in topological insulators,” Nanomater. and Nanotecho. 4(13), 1–8 (2014).
[Crossref]

Nat. Commun. (1)

Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, Z. Lin, D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong, L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang, Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou, “Orbital-selective spin texture and its manipulation in a topological insulator,” Nat. Commun. 5, 1–9 (2014).
[Crossref]

Nat. Nanotechnol. (1)

P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, “Observation of Dirac plasmons in a topological insulator,” Nat. Nanotechnol. 8, 556–560 (2013).
[Crossref] [PubMed]

Nat. Phys. (1)

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, 438–442 (2009).
[Crossref]

Nature Commun. (1)

J.-Y. Ou, J.-K. So, G. Adamo, A. Sulaev, L. Wang, and N. I. Zheludev, “Ultraviolet and visible range plasmonics in the topological insulator Bi1.5Sb0.5Te1.8Se1.2,” Nature Commun. 55139 (2014).
[Crossref]

Phys. Rev. B (7)

P. Di Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi, “Optical conductivity of bismuth-based topological insulators,” Phys. Rev. B 86, 045439 (2012).
[Crossref]

S. Sim, M. Brahlek, N. Koirala, S. Cha, S. Oh, and H. Choi, “Ultrafast terahertz dynamics of hot Dirac-electron surface scattering in the topological insulator Bi2Se3,” Phys. Rev. B 89, 165137 (2014).
[Crossref]

S. Sim, N. Koirala, M. Brahlek, J. H. Sung, J. Park, S. Cha, M.-H. Jo, S. Oh, and H. Choi, “Tunable Fano quantum-interference dynamics using a topological phase transition in (Bi1x Inx)2Se3,” Phys. Rev. B 91, 235438 (2015).
[Crossref]

T. Stauber, G. Gomez-Santós, and L. Brey, “Spin-charge separation of plasmonic excitations in thin topological insulators,” Phys. Rev. B 88(20), 205427 (2013).
[Crossref]

M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61(12), 8187–8201 (2000).
[Crossref]

R. E. V. Profumo, R. Asgari, M. Polini, and A. H. MacDonald, “Double-layer graphene and topological insulator thin-film plasmons,” Phys. Rev. B 85(8), 085443 (2012).
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A. Deneuville, D. Tanner, and P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43(8), 6544–6550 (1991).
[Crossref]

Phys. Rev. Lett. (7)

V. R. 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(8), 087403 (2012).
[Crossref]

S. Das Sarma and E. H. Hwang, “Collective modes of the massless Dirac plasma,” Phys. Rev. Lett. 102(20), 206412 (2009).
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L. Fu, C. L. Kane, and E. J. Mele, “Topological insulators in three dimensions,” Phys. Rev. Lett. 98(10), 106803 (2007).
[Crossref] [PubMed]

O. V. Yazyev, J. E. Moore, and S. G. Louie, “Spin polarization and transport of surface states in the topological insulators Bi2Se3 and Bi2Te3 from first principles,” Phys. Rev. Lett. 105(26), 266806 (2010).
[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-Diraccone topological-insulator states in Bi2Te3 and Sb2Te3,” Phys. Rev. Lett. 103(14), 146401 (2009).
[Crossref] [PubMed]

Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R. Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik, “Measurement of intrinsic Dirac fermion cooling on the surface of the topological insulator Bi2Se3 using time-resolved and angle-resolved photoemission spectroscopy,” Phys. Rev. Lett. 109(12), 127401 (2012).
[Crossref] [PubMed]

S. Raghu, S. B. Chung, X. L. Qi, and S. C. Zhang, “Collective modes of a helical liquid,” Phys. Rev. Lett. 104(11), 116401 (2010)
[Crossref] [PubMed]

Phys. Stat. Sol. (b) (1)

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Phys. Stat. Sol.(a) (1)

El. H. Kaddouri, T. Maurice, X. Gratens, S. Charar, S. Benet, A. Mefleh, J. C. Tedenac, and B. Liautard, “Optical properties of bismuth telluride thin films, Bi2Te3/Si(100) and Bi2Te)3/SiO2/Si(100),” Phys. Stat. Sol.(a) 176(2), 1071–1076 (1999).
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A. Yu. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys. Usp. 44(10S), 131 (2001).
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Rev. Mod. Phys. (2)

M. Z. Hasan and C. L. Kane, “Colloquium : topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Other (5)

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), 178–181 (2009).
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I. Appelbaum, H. D. Drew, and M. S. Fuhrer, “Proposal for a topological plasmon spin rectifier,” http://arxiv.org/abs/1011.2697

S. A. Maier, Plasmonics: Fundamentals and Applications(Springer, 2007).

A. L. Yeats, Y. Pan, A. Richardella, P. J. Mintun, N. Samarth, and D. D. Awschalom, “Persistent optical gating of a topological insulator,” Sci. Adv.1 (2014).

M.S. Wolf, Infrared and Optical Studies of Topological Insulators Bi2Se3, Bi2Te3 and Sb2Te3, (Thesis, The University of Akron, 2011).

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Figures (8)

Fig. 1
Fig. 1

Schematics of the experiment for observing SPP resonances in Bi2Se3 (see [22]). Thin film, MBE grown on sapphire, is etched into a periodic array of ribbons of the width W, separated by a gap of the same width. Incident linearly polarized far-IR light excites standing waves of collective oscillations of surface charges. The resonant excitation for given ribbon width W is observed in the drop of the transmitted light intensity T(ν). The surface current density j, plotted on the vertical axis, describes standing waves with the period 2W.

Fig. 2
Fig. 2

Contributions of various terms of the Drude-Lorentz model into the real part of the far-IR bulk dielectric function of Bi2Se3. Two IR-active phonon modes (Lorentz-α with ω ≈ 61 cm1 and Lorentz-β with ω ≈ 133 cm1) correspond to the first two terms in the Drude-Lorentz model (3). The oscillator Lorentz-Ω accounts for higher frequency absorption, in this case at ω ≈ 2029.5 cm1 = 252 meV which is close to the band-gap energy of Bi2Se3 bulk [28].

Fig. 3
Fig. 3

Real ε′ and imaginary ε″ parts of the bulk dielectric function of Bi2Se3, according to the Drude-Lorentz model. Solid lines show the bulk dielectric function without the Drude term, e.g. in a perfect crystal at low temperature. Dashed lines reproduce the bulk dielectric function with the Drude term present. The effect of Drude term is very strong in this spectral range.

Fig. 4
Fig. 4

(a, b) Dispersion relations of SPPs in Bi2Se3 films of several thicknesses for (a) kF = 0.15 × 109 m1 (EF = 50 meV) and (b) kF = 1.52 × 109 m1 (EF = 500 meV). Cirlce, square and triangle marks represent the experimental values from [22]. (c, d) Propagation lengths of SPPs in Bi2Se3 films for corresponding parameters: (c) for kF = 0.15 × 109 m1, (d) for kF = 1.52 × 109 m1. The lines marked α ≈ 63 cm1 and β ≈ 133 cm1 indicate positions of IR-active modes interacting with SPPs and causing non-monotinic behavior of dispersion and propagation length.

Fig. 5
Fig. 5

Dispersion curves of SPPs in Bi2Se3 thin films, calculated using the Drude-Lorentz model and the parameters reported in [22] (kF = 1.37 × 109 m1, EF = 541 meV). Cirlce, square and triangle marks correspond to the experimental results [22].

Fig. 6
Fig. 6

Propagation lengths of SPPs in Bi2Se3 thin films, calculated using the Drude-Lorentz model and the parameters reported in [22] (kF = 1.37 × 109 m1, EF = 541 meV).

Fig. 7
Fig. 7

(a, b) Dispersion relations of SPPs in Bi2Te3 films of several thicknesses for (a) kF = 0.15 × 109 m1 (EF = 50 meV) and (b) kF = 1.52 × 109 m1 (EF = 500 meV). (c, d) Propagation lengths of SPPs in Bi2Te3 films for corresponding parameters: (c) for kF = 0.15 × 109 m1, (d) for kF = 1.52 × 109 m1.

Fig. 8
Fig. 8

Tuning SPP dispersion by varying the number of units cells in superlattice of TI and dielectric (ZnSe in this case). Transfer matrix calculations for the structure reported in [36].

Tables (5)

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Table 1 Parameters for the Drude-Lorentz model with ε = 1, extracted from the reflectance measurements on Bi2Se3 at room temperatures in far and mid-IR [27].

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Table 2 Parameters for the Drude-Lorentz model for the unpatterned Bi2Se3 thin films at 300 K, see Supplementary Material to [22].

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Table 3 Parameters of the Drude-Lorentz model with ε = 85, extracted from the reflectance measurements on Bi2Te3 at room temperatures in far-IR [28].

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Table 4 Parameters for the Drude-Lorentz model with ε = 1 extracted from the reflectance measurements on Bi2Te3 at room temperatures in far- and mid-IR [27].

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Table 5 Parameters for the Drude-Lorentz model with ε = 51 extracted from the reflectance measurements on Sb2Te3 at room temperatures in far and mid-IR [27, 28].

Equations (4)

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ω 2 = v F k F e 2 ε 0 h q ε T + ε B + q d ε T I
ε ( ω , cm 1 ) = n 0 2 + ( n 0 2 1 ) ( λ ω ) 2 + i γ ( n 0 2 1 ) ( λ ω ) ,
ε ( ω , cm 1 ) = ε ω D 2 ω 2 + i ω γ D + j = 1 j = 3 , 4 ω p j 2 ω 0 j 2 ω 2 i ω γ j .
q ( ω ) = A ω 2 [ ε T ( ω ) + ε B ( ω ) ] 1 A ω 2 d ε T I ( ω ) , A ε 0 h ν F k F e 2 .

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