Abstract

Silicene is a competitive and promising 2D material, possessing interesting topological, electronic and optical properties. The presence of strong spin orbit interaction in silicene and its analogues, germanene and tinene, leads to the opening of a gapin the energy spectrum and spin-splitting of the bands in each valley. Building upon prior work we discuss a general method to determine the magneto-optic response of silicene when a Gaussian beam is incident on silicene grown on a dielectric substrate in the presence of a static magnetic field. We use a semiclassical treatment to describe the Faraday rotation (FR) and Magneto-optical Kerr effect (MOKE). The response can be modulated both electrically and magnetically. We derive analytic expressions for valley and spin polarized FR and MOKE for arbitrary polarization of incident light in the terahertz regime. We demonstrate that large FR and MOKE can be achieved by tuning the electric field, magnetic fields and chemical potential in these fascinating 2D materials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]

2019 (3)

G. Catarina, J. Have, J. Fernández-Rossier, and N. M. R. Peres, “Optical orientation with linearly polarized light in transition metal dichalcogenides,” Phys. Rev. B 99(12), 125405 (2019).
[Crossref]

A. Dolatabady and N. Granpayeh, “Manipulation of the Faraday rotation by graphene metasurfaces,” J. Magn. Magn. Mater. 469, 231-235 (2019).
[Crossref]

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
[Crossref]

2018 (1)

T. P. Cysne, T. G. Rappoport, J. H. Garcia, and A. R. Rocha, “Quantum Hall effect in graphene with interface-induced spin-orbit coupling,” Phys. Rev. B 97(8), 085413 (2018).
[Crossref]

2017 (4)

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

J. M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. Commun. 8, 14626 (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]

M. Oliva-Leyva and C. Wang, “Magneto-optical conductivity of anisotropic two-dimensional Dirac-Weyl materials,” Annals of Physics 384, 61–70 (2017).
[Crossref]

2016 (2)

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
[Crossref] [PubMed]

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
[Crossref]

2015 (4)

S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
[Crossref] [PubMed]

H. Funk, A. Knorr, F. Wendler, and E. Malic, “Microscopic view on Landau level broadening mechanisms in graphene,” Phys. Rev. B 92(20), 205428 (2015).
[Crossref]

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
[Crossref]

2014 (7)

T. Low, R. Roldan, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014).
[Crossref] [PubMed]

T. Low, R. Roldan, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014).
[Crossref]

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[Crossref] [PubMed]

M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
[Crossref] [PubMed]

A. Pospischil, M. M. Furchi, and T. Mueller, “Solar-energy conversion and light emission in an atomic monolayer p-n diode,” Nat. Nanotechnol. 9, 257–261 (2014).
[Crossref] [PubMed]

K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, “The valley Hall effect in MoS2 transistors,” Science 344(6191), 1489–1492 (2014).
[Crossref] [PubMed]

M. Lasia and L. Brey, “Optical properties of magnetically doped ultrathin topological insulator slabs,” Phys. Rev. B 90(7), 075417 (2014).
[Crossref]

2013 (10)

C. J. Tabert and E. J. Nicol, “Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals,” Phys. Rev. Lett. 110(19), 197402 (2013).
[Crossref] [PubMed]

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for temperature-dependence of the Faraday effect by optical activity temperature shift,” IEEE Sens. J. 13(10), 832–837 (2013).
[Crossref]

M. Tahir and U. Schwingenschlögl, “Valley polarized quantum Hall effect and topological insulator phase transitions in silicene,” Sci. Rep. 3, 1075 (2013).
[Crossref] [PubMed]

C. J. Tabert and E. J. Nicol, “Magneto-optical conductivity of silicene and other buckled honeycomb lattices,” Phys. Rev. B 88(8), 085434 (2013).
[Crossref]

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

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

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
[Crossref] [PubMed]

T. Yoshino, “Theory for oblique-incidence magneto-optical Faraday and Kerr effects in interfaced monolayer graphene and their characteristic features,” J. Opt. Soc. Am. B 30(5), 1085–1091 (2013).
[Crossref]

M. Tymchenko, A. Y. Nikitin, and L. Martn-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry-Pérot enhanced faraday rotation in graphene,” Opt. Express 21(21), 24736–24741 (2013).
[Crossref] [PubMed]

2012 (5)

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

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

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6 (11), 749–758 (2012).
[Crossref]

M. Ezawa, “Spin-valley optical selection rule and strong circular dichroism in silicene,” Phys. Rev. B 86(16), 161407 (2012).
[Crossref]

2011 (5)

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

F. H. Koppens, D. E. Chang, and F. Javier, “Graphene plasmonics: A platform for strong light–matter interactions,” Nano Lett,  11(8), 3370–3377 (2011).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84(23), 235410 (2011).
[Crossref]

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98(2), 021911 (2011).
[Crossref]

2009 (1)

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]

2008 (1)

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

2007 (3)

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, “Electronic spin transport and spin precession in single graphene layers at room temperature,” Nature,  448, 571–574 (2007).
[Crossref] [PubMed]

G. G. Guzmán-Verri and L. C. Lew Yan Voon, “Electronic structure of silicon-based nanostructures,” Phys. Rev. B 76(7), 075131 (2007).
[Crossref]

2006 (1)

V. P. Gusynin and S. G. Sharapov, “Transport of Dirac quasiparticles in graphene: Hall and optical conductivities,” Phys. Rev. B 73(24), 245411 (2006).
[Crossref]

2005 (1)

V. P. Gusynin and S. G. Sharapov, “Unconventional integer quantum Hall effect in graphene,” Phys. Rev. Lett. 95(14), 146801 (2005).
[Crossref] [PubMed]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

2003 (1)

1995 (1)

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7(12), 1468–1470 (1995).
[Crossref]

Ahmed, S.

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

Akashi, R.

R. Suzuki, M. Sakano, Y. J. Zhang, R. Akashi, D. Morikawa, A. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto, T. Okuda, K. Ishizaka, R. Arita, and Y. Iwasa, “Valley-dependent spin polarization in bulk MoS2 with broken inversion symmetry,” Nat. Nanotechnol. 9, 611–617 (2014).
[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]

Amorim, B.

W. J. M. Kort-Kamp, B. Amorim, G. Bastos, F. A. Pinheiro, F. S. S. Rosa, N. M. R. Peres, and C. Farina, “Active magneto-optical control of spontaneous emission in graphene,” Phys. Rev. B 92(20), 205415 (2015).
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Arita, R.

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J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
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S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520, 69–72 (2015).
[Crossref] [PubMed]

D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
[Crossref] [PubMed]

H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

W. Yao, D. Xiao, and Q. Niu, “Valley-dependent optoelectronics from inversion symmetry breaking,” Phys. Rev. B 77(23), 235406 (2008).
[Crossref]

D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
[Crossref]

Yao, Y. G.

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

Yi, J.

S. Ahmed and J. Yi, “Two-dimensional transition metal dichalcogenides and their charge carrier mobilities in field-effect transistors,” Nano-Micro Lett. 9, 50 (2017).
[Crossref]

Yokota, M.

Yoshimi, R.

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
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Yoshino, T.

Yu, H.

J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
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Zaitsev, A. D.

A. N. Grebenchukov, S. E. Azbite, A. D. Zaitsev, and M. K. Khodzitsky, “Faraday effect control in graphene-dielectric structure by optical pumping,” J. Magn. Magn. Mater. 472, 25–28 (2019).
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B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
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H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, “Valley polarization in MoS2 monolayers by optical pumping,” Nat. Nanotech. 7(8), 490 (2012).
[Crossref]

Zhang, H.-J.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
[Crossref] [PubMed]

Zhang, S.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
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Zhang, S. C.

Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
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Zhang, W.

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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Zhang, X.A.

Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
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Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
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Zou, Y.

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
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Q. Yang, X.A. Zhang, A. Bagal, W. Guo, and C.H. Chang, “Antireflection effects at nanostructured material interfaces and the suppression of thin-film interference,” Nanotechnology 24(23), 235202 (2013).
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Nat. Commun. (2)

K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, “Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state,” Nat. Commun. 7, 12245 (2016).
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J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. 1, 16055 (2016).
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Opt. Express (1)

Phys. Chem. Chem. Phys. (1)

B. Cai, S. Zhang, Z. Hu, Y. Hu, Y. Zou, and H. Zeng, “Tinene: a two-dimensional Dirac material with a 72 meV band gap,” Phys. Chem. Chem. Phys. 17(19), 12634–12638 (2015).
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Phys. Rev. A (1)

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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D. Xiao, G. B. Liu, W. Feng, X. Xu, and W. Yao, “Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides,” Phys. Rev. Lett. 108(19), 196802 (2012).
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D. Xiao, W. Yao, and Q. Niu, “Valley-contrasting physics in graphene: Magnetic moment and topological transport,” Phys. Rev. Lett. 99(23), 236809 (2007).
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M. A. Cazalilla, H. Ochoa, and F. Guinea, “Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides,” Phys. Rev. Lett. 113(7), 077201 (2014).
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C. C. Liu, W. Feng, and Y. G. Yao, “Quantum spin Hall effect in silicene and two-dimensional germanium,” Phys. Rev. Lett. 107(7), 076802 (2011).
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Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S. C. Zhang, “Large-gap quantum spin Hall insulators in tin films,” Phys. Rev. Lett. 111(13), 136804 (2013).
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Figures (6)

Fig. 1
Fig. 1 Faraday, Kerr rotation and ellipticity of silicene-substrate system as function of photon energy electric and magnetic fields. (a) The s polarized Faraday rotation and (b) ellipticity as function of incident photon energy in the K valley with modulation of the external electric field for the three distinct topological regimes, TI, VSPM and BI for a magnetic field of 1 T. The spectral peaks are labelled 1 through 6 and their origin is identified in the main text. The spectrum are vertically shifted by 15° among themselves for clearer viewing. Furthermore, in this figure we use Δz = Δso/2 (TI) and Δz = 2Δso (BI). (c) The s polarized Kerr rotation as function of incident photon energy in the Kʹ valley with modulation of the external magnetic field for the TI regime for three different values of B = 1, 3 and 5 T. (d) The maximum Faraday, Kerr rotation and ellipticity as function of magnetic field in K valley for the single transition Δ−10,K,↑. The parameters used are θ1 = 30°, Γ = 0.01Δso, refractive index n2 = 1.84 and chemical potential μF = 0.
Fig. 2
Fig. 2 (a) Schematic representation of the allowed transitions between LL’s for three different values of chemical potential μF = 0, 10 and 22 meV; (b) and (c) the s polarized Kerr rotation as function of incident photon frequency in K and valleys with modulation of the chemical potential in the TI regime for a magnetic field of 1 T, respectively. (d) The s polarized Faraday rotation as function of incident photon energy in K valley for different incident angles for a single transition in the TI regime. (e) The s polarized Faraday rotation as function of incident photon energy in K for different temperatures for a single transition in the TI regime. (f) The s and p polarized Kerr rotation as function of incident photon frequency in the semiclassical limit for n-type and p-type silicene (μF = 56 and -56 meV), respectively. The solid line represents the s polarized and the dashed line p polarized. The parameters used are θ = 30°, Γ = 0.01Δso and refractive index n2 = 1.84.
Fig. 3
Fig. 3 (a) Schematic representation of the allowed transitions between LL’s for chemical potential μ = 0.2. (b) and (c) the s polarized Faraday rotation contour plots as function of x in K valley for μ = 0 and 1.25, repectively, where x = Δzso and μ = μFso. The parameters used are θ = 30°, Γ = 0.01Δso and refractive index n2 = 1.84.
Fig. 4
Fig. 4 Faraday and Kerr rotation of silicene-substrate system as function of photon energy for three different relative permittivities. (a) The s polarized and (b) p polarized Faraday rotation as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr rotation as a function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso and chemical potential μF = 0.
Fig. 5
Fig. 5 Faraday and Kerr rotation of silicene-substrate system as a function of photon energy for three different relative permeabilitis. (a) The s polarized and (b) p polarized Faraday rotation as a function of incident photon energy in the K valley with modulation of relative permeabilitis in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr rotation as a function of incident photon energy in the K valley with modulation of relative permeabilities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso, refractive index n2 = 1.84 and chemical potential μF = 0.
Fig. 6
Fig. 6 Faraday and Kerr ellipticities as a function of incident photon energy for three different relative permittivities. (a) The s polarized and (b) p polarized Faraday ellipticities as function of incident photon energy in the K valley with modulation of relative permittivities in the TI regime for a magnetic field of 1 T. (c) The s polarized and (d) p polarized Kerr ellipticities as a function of incident photon energy in the K valley with relative permittivities in the TI regime for a magnetic field of 1 T, for the two transitions. The parameters used are θ1 = 30°, Γ = 0.01Δso and chemical potential μF = 0.

Tables (1)

Tables Icon

Table 1 Table of allowed transitions in K valley in the n = −1,0,1 subspace, at a fixed Magnetic field and Chemical potential μ. Furthermore x = Δzso, y = v 2 e B / Δ s o 2 . and μ = μFso.

Equations (29)

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H ^ ξ σ = v F ( ξ k x τ ^ x + k y τ ^ y ) 1 2 ξ Δ s o σ ^ z τ ^ z + 1 2 Δ z τ ^ z
E ( ξ , σ , n , t ) = { t 2 v F 2 e B | n | + Δ ξ σ 2 , if n 0. ξ Δ ξ σ , if n = 0.
| n ¯ | ξ = 1 = ( i A n | n 1 B n | n )
| n ¯ | ξ = 1 = ( i A n | n B n | n 1 )
A n = { | E ( ξ , σ , n , t ) | + t Δ ξ σ 2 | E ( ξ , σ , n , t ) | , if n 0. 1 ξ 2 , if n = 0.
B n = { | E ( ξ , σ , n , t ) | t Δ ξ σ 2 | E ( ξ , σ , n , t ) | , if n 0. 1 + ξ 2 , if n = 0.
σ μ ν ( Ω ) = i 2 π l B 2 σ , ξ = ± 1 m n f n f m E n E m n ¯ | j ^ μ | m ¯ m ¯ | j ^ ν | n ¯ Ω ( E n E m ) + i Γ ,
R e I m } ( σ x x ( Ω ) ) σ 0 = 2 v 2 e B π ξ , σ m , n Θ ( E n μ F ) Θ ( E m μ F ) E n E m × [ ( A m B n ) 2 δ | m | ξ , | n | + ( B m A n ) 2 δ | m | + ξ , | n | ] { F G ,
R e I m } ( σ x y ( Ω ) ) σ 0 = 2 v 2 e B π ξ , σ m , n ξ Θ ( E n μ F ) Θ ( E m μ F ) E n E m × [ ( A m B n ) 2 δ | m | ξ , | n | ( B m A n ) 2 δ | m | + ξ , | n | ] { G F .
r p p = α + T α L + β α + T α + L + β ,
r s s = ( α T α + L + β α + T α + L + β ) ,
t p p = 2 Z 2 ε 2 Z 1 k 1 z α + T α + T α + L + β ,
t s s = 2 μ 2 k 1 z α + L α + T α + L + β ,
r s p = t s p = 2 Z 0 2 μ 0 μ 1 μ 2 k 1 z k 2 z ( σ H + σ x y s y m ) Z 1 ( α + T α + L + β ) ,
r p s = k 1 k 2 z k 2 k 1 z t p s = 2 Z 0 2 μ 1 μ 2 Z 1 k 1 z k 2 z ( σ x y s y m σ H ) α + T α + L + β ,
α ± L = ( k 1 z ε 2 ± k 2 z ε 1 + k 1 z k 2 z σ L / ( ε 0 Ω ) ) ,
α ± T = ( k 2 z μ 1 ± k 1 z μ 2 + μ 0 μ 1 μ 2 σ T Ω ) ,
β = Z 0 2 μ 1 μ 2 k 1 z k 2 z [ σ H 2 ( σ x y s y m ) 2 ]
Θ F , s ( p ) = 1 2 tan 1 ( 2 Re ( χ F , s ( p ) ) 1 | χ F , s ( p ) | 2 ) ,
and η F , s ( p ) = 1 2 sin 1 ( 2 Im ( χ F , s ( p ) ) 1 | χ F , s ( p ) | 2 ) ,
χ F , s = t p s t s s = Z 0 ε 1 μ 1 k 1 c o s ( θ 1 ) σ H α + L ,
and χ F , p = t s p t p p = Z 0 μ 2 ε 2 μ 0 μ 1 k 2 c o s ( θ 2 ) σ H α + T
Θ K , s ( p ) = 1 2 tan 1 ( 2 Re ( χ K , s ( p ) ) 1 | χ K , s ( p ) | 2 ) ,
and η K , s ( p ) = 1 2 sin 1 ( 2 Im ( χ K , s ( p ) ) 1 | χ K , s ( p ) | 2 ) ,
χ K , s = r p s r s s = 2 Z 0 μ 1 ε 1 μ 2 k 1 z k 2 z σ H α T α + L + β ,
and χ K , p = r s p r p p = 2 Z 0 μ 1 ε 1 μ 0 μ 2 k 1 z k 2 z σ H α + L α L + β
E n + 1 E n v 2 e B Δ ξ σ 2 + 2 n v 2 e B = Ω c ,
Re ( σ x x ( Ω ) ) σ 0 = Im ( σ x y ( Ω ) ) σ 0 = μ F π Γ ( ( Ω Ω c ) ) 2 + Γ 2 ,
Im ( σ x x ( Ω ) ) σ 0 = Re ( σ x y ( Ω ) ) σ 0 = μ F π ( Ω Ω c ) ( ( Ω Ω c ) ) 2 + Γ 2

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