Abstract

We illustrate Kerr and Faraday rotation in the strained-graphene by applying the second quantization method as an alternative approach. We consider the right- and left-going photon fields coupling with strained graphene. In other words, we have a new stationary state solution describing this phenomenon. A single-photon polarization in the provided state is considered in cases of a non-magnetic field, and uniform strained graphene. We show that the optical l properties of Faraday rotation, reflectance, and transmittance depend on the spinor phase and the energy level of an electron in strained graphene. These values can be controlled by variation of a strain parameter and strain types. Then, it is possible to have an alternative measurement of the pseudo-spin state and electronic structure in the 2-D layer by observing the optical properties of the single-photon in the provided state.

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

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  5. A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).
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    [Crossref]
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    [Crossref]
  21. S-J. Zhang, H. Pan, and H-L. Wang, “Frequency dependent optical conductivity of strained graphene at T=0 from an effective quantum field theory,” Physica B 511, 80 (2017).
    [Crossref]
  22. H. P. Paudel and M. N. Leuenberger, “Giant Faraday effect due to Pauli exclusion principle in 3D topological insulators,” J. Phys.: Condens. Matter 26, 082201 (2014).
  23. T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
    [Crossref]
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    [Crossref]
  25. P. Dietl, F. Piechon, and G. Montambaux, “New magnetic field dependence of landau levels in a graphenelike structure,” Phys. Rev. lett. 100, 235405 (2008).
    [Crossref]
  26. T. Thitapura, W. Liewrian, T. Jutarosaga, and S. Boonchui, “Curvature effect on polarization of light emitted from chiral carbon nanotubes,” Opt. Express 25, 25588 (2017).
    [Crossref] [PubMed]

2018 (1)

D. Georgiev and E. Cohen, “Probing finite coarse-grained virtual Feynman histories with sequential weak values,” Phy. Rev. A 97, 052102 (2018).
[Crossref]

2017 (4)

S-J. Zhang, H. Pan, and H-L. Wang, “Frequency dependent optical conductivity of strained graphene at T=0 from an effective quantum field theory,” Physica B 511, 80 (2017).
[Crossref]

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

T. Thitapura, W. Liewrian, T. Jutarosaga, and S. Boonchui, “Curvature effect on polarization of light emitted from chiral carbon nanotubes,” Opt. Express 25, 25588 (2017).
[Crossref] [PubMed]

Z. B. Siu, S. G. Tan, and M. B. A. Jalil, “Effective Hamiltonian for surface states of topological insulator nanotubes,” Sci. Rep. 7, 45350 (2017).
[Crossref] [PubMed]

2016 (1)

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]

2015 (2)

P. Srisangyingcharoen, R. Klinkla, and S. Boonchui, “Dynamics of propagating surface plasmon induced photon emission from quantum dots: quantum history approach,” J. Phys. B: At. Mol. Opt. Phys. 48, 215501 (2015).
[Crossref]

P. Fanbanrai, A. Hutema, and S. Boonchui, “Effects of strain on the Schwinger pair creation in graphene,” Physica B 472, 84 (2015).
[Crossref]

2014 (3)

H. P. Paudel and M. N. Leuenberger, “Giant Faraday effect due to Pauli exclusion principle in 3D topological insulators,” J. Phys.: Condens. Matter 26, 082201 (2014).

M. Laakso and M. Pletyukhov, “Scattering of two photons from two distant qubits: exact solution,” Phys. Rev. lett. 113, 183601 (2014).
[Crossref] [PubMed]

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

2013 (1)

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

2012 (2)

A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg, “Lattice-corrected strain-induced vector potentials in graphene,” Phys. Rev. B 85, 159909 (2012).
[Crossref]

J. C. Martinez, M. B. A. Jalil, and S. G. Tan, “Giant Faraday and Kerr rotation with strained graphene,” Opt. Lett. 37, 3237 (2012)
[Crossref] [PubMed]

2011 (1)

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

2010 (4)

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

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

N. M. R. Peres, “The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82, 2673 (2010).
[Crossref]

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

2009 (2)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

V. M. Pereira and A. H. C. Neto, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80, 045401 (2009).
[Crossref]

2008 (1)

P. Dietl, F. Piechon, and G. Montambaux, “New magnetic field dependence of landau levels in a graphenelike structure,” Phys. Rev. lett. 100, 235405 (2008).
[Crossref]

2007 (1)

D. E. Chang, A. S. Sorensen, A. E. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807 (2007).
[Crossref]

2006 (1)

B. A. Bernevig, T. L. Hughes, and S. C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757 (2006).
[Crossref] [PubMed]

2005 (1)

1997 (1)

R. O. Dillon, I. L. Spain, and J. W. McClure, “Electronic energy band parameters of graphite and their dependence on pressure, temperature and acceptor concentration,” J. Phys. Chem. Solids. 38, 635 (1997).
[Crossref]

Aoki, H.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[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, 1757 (2006).
[Crossref] [PubMed]

Bludov, Y. V.

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

Boonchui, S.

T. Thitapura, W. Liewrian, T. Jutarosaga, and S. Boonchui, “Curvature effect on polarization of light emitted from chiral carbon nanotubes,” Opt. Express 25, 25588 (2017).
[Crossref] [PubMed]

P. Fanbanrai, A. Hutema, and S. Boonchui, “Effects of strain on the Schwinger pair creation in graphene,” Physica B 472, 84 (2015).
[Crossref]

P. Srisangyingcharoen, R. Klinkla, and S. Boonchui, “Dynamics of propagating surface plasmon induced photon emission from quantum dots: quantum history approach,” J. Phys. B: At. Mol. Opt. Phys. 48, 215501 (2015).
[Crossref]

Bormann, R.

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

Bostwick, A.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Chang, D. E.

D. E. Chang, A. S. Sorensen, A. E. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807 (2007).
[Crossref]

Chaves, A. J.

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

Cohen, E.

D. Georgiev and E. Cohen, “Probing finite coarse-grained virtual Feynman histories with sequential weak values,” Phy. Rev. A 97, 052102 (2018).
[Crossref]

Crassee, I.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Dai, X.

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

Danz, T.

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

de Paula, W.

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

Demler, A. E.

D. E. Chang, A. S. Sorensen, A. E. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807 (2007).
[Crossref]

Dietl, P.

P. Dietl, F. Piechon, and G. Montambaux, “New magnetic field dependence of landau levels in a graphenelike structure,” Phys. Rev. lett. 100, 235405 (2008).
[Crossref]

Dillon, R. O.

R. O. Dillon, I. L. Spain, and J. W. McClure, “Electronic energy band parameters of graphite and their dependence on pressure, temperature and acceptor concentration,” J. Phys. Chem. Solids. 38, 635 (1997).
[Crossref]

Fan, S.

Fanbanrai, P.

P. Fanbanrai, A. Hutema, and S. Boonchui, “Effects of strain on the Schwinger pair creation in graphene,” Physica B 472, 84 (2015).
[Crossref]

Fang, Z.

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

Ferreira, A.

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

Frederico, T.

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

Gaida, H. J.

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Georgiev, D.

D. Georgiev and E. Cohen, “Probing finite coarse-grained virtual Feynman histories with sequential weak values,” Phy. Rev. A 97, 052102 (2018).
[Crossref]

Goldberg, B. B.

A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg, “Lattice-corrected strain-induced vector potentials in graphene,” Phys. Rev. B 85, 159909 (2012).
[Crossref]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Hibino, H.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

Hughes, T. L.

B. A. Bernevig, T. L. Hughes, and S. C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757 (2006).
[Crossref] [PubMed]

Hutema, A.

P. Fanbanrai, A. Hutema, and S. Boonchui, “Effects of strain on the Schwinger pair creation in graphene,” Physica B 472, 84 (2015).
[Crossref]

Jalil, M. B. A.

Z. B. Siu, S. G. Tan, and M. B. A. Jalil, “Effective Hamiltonian for surface states of topological insulator nanotubes,” Sci. Rep. 7, 45350 (2017).
[Crossref] [PubMed]

J. C. Martinez, M. B. A. Jalil, and S. G. Tan, “Giant Faraday and Kerr rotation with strained graphene,” Opt. Lett. 37, 3237 (2012)
[Crossref] [PubMed]

Jutarosaga, T.

Kawasaki, M.

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]

Kitt, A. L.

A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg, “Lattice-corrected strain-induced vector potentials in graphene,” Phys. Rev. B 85, 159909 (2012).
[Crossref]

Klinkla, R.

P. Srisangyingcharoen, R. Klinkla, and S. Boonchui, “Dynamics of propagating surface plasmon induced photon emission from quantum dots: quantum history approach,” J. Phys. B: At. Mol. Opt. Phys. 48, 215501 (2015).
[Crossref]

Kuzmenko, A. B.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Laakso, M.

M. Laakso and M. Pletyukhov, “Scattering of two photons from two distant qubits: exact solution,” Phys. Rev. lett. 113, 183601 (2014).
[Crossref] [PubMed]

Leuenberger, M. N.

H. P. Paudel and M. N. Leuenberger, “Giant Faraday effect due to Pauli exclusion principle in 3D topological insulators,” J. Phys.: Condens. Matter 26, 082201 (2014).

Levallois, J.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Liewrian, W.

Liu, C-X.

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

Lukin, M. D.

D. E. Chang, A. S. Sorensen, A. E. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807 (2007).
[Crossref]

MacDonald, A. H.

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

Martinez, J. C.

Matsunaga, R.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

McClure, J. W.

R. O. Dillon, I. L. Spain, and J. W. McClure, “Electronic energy band parameters of graphite and their dependence on pressure, temperature and acceptor concentration,” J. Phys. Chem. Solids. 38, 635 (1997).
[Crossref]

Mogi, M.

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]

Montambaux, G.

P. Dietl, F. Piechon, and G. Montambaux, “New magnetic field dependence of landau levels in a graphenelike structure,” Phys. Rev. lett. 100, 235405 (2008).
[Crossref]

Morimoto, T.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

Neff, A.

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

Neto, A. H. C.

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

V. M. Pereira and A. H. C. Neto, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80, 045401 (2009).
[Crossref]

Novoselov, K. S.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Ogawa, N.

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]

Okada, K. N.

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]

Oliveira, O.

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

Ostler, M.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Pan, H.

S-J. Zhang, H. Pan, and H-L. Wang, “Frequency dependent optical conductivity of strained graphene at T=0 from an effective quantum field theory,” Physica B 511, 80 (2017).
[Crossref]

Paudel, H. P.

H. P. Paudel and M. N. Leuenberger, “Giant Faraday effect due to Pauli exclusion principle in 3D topological insulators,” J. Phys.: Condens. Matter 26, 082201 (2014).

Pereira, V.

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

Pereira, V. M.

A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg, “Lattice-corrected strain-induced vector potentials in graphene,” Phys. Rev. B 85, 159909 (2012).
[Crossref]

V. M. Pereira and A. H. C. Neto, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80, 045401 (2009).
[Crossref]

Peres, N. M. R.

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

N. M. R. Peres, “The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82, 2673 (2010).
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Piechon, F.

P. Dietl, F. Piechon, and G. Montambaux, “New magnetic field dependence of landau levels in a graphenelike structure,” Phys. Rev. lett. 100, 235405 (2008).
[Crossref]

Pletyukhov, M.

M. Laakso and M. Pletyukhov, “Scattering of two photons from two distant qubits: exact solution,” Phys. Rev. lett. 113, 183601 (2014).
[Crossref] [PubMed]

Qi, X-L.

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

Ropers, C.

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

Rotenberg, E.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Santos, M. C.

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

Schäfer, S.

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

Seyller, T.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Shen, J. T.

Shimano, R.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

Siu, Z. B.

Z. B. Siu, S. G. Tan, and M. B. A. Jalil, “Effective Hamiltonian for surface states of topological insulator nanotubes,” Sci. Rep. 7, 45350 (2017).
[Crossref] [PubMed]

Sorensen, A. S.

D. E. Chang, A. S. Sorensen, A. E. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807 (2007).
[Crossref]

Spain, I. L.

R. O. Dillon, I. L. Spain, and J. W. McClure, “Electronic energy band parameters of graphite and their dependence on pressure, temperature and acceptor concentration,” J. Phys. Chem. Solids. 38, 635 (1997).
[Crossref]

Srisangyingcharoen, P.

P. Srisangyingcharoen, R. Klinkla, and S. Boonchui, “Dynamics of propagating surface plasmon induced photon emission from quantum dots: quantum history approach,” J. Phys. B: At. Mol. Opt. Phys. 48, 215501 (2015).
[Crossref]

Swan, A. K.

A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg, “Lattice-corrected strain-induced vector potentials in graphene,” Phys. Rev. B 85, 159909 (2012).
[Crossref]

Takahashi, K. S.

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]

Takahashi, Y.

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]

Tan, S. G.

Z. B. Siu, S. G. Tan, and M. B. A. Jalil, “Effective Hamiltonian for surface states of topological insulator nanotubes,” Sci. Rep. 7, 45350 (2017).
[Crossref] [PubMed]

J. C. Martinez, M. B. A. Jalil, and S. G. Tan, “Giant Faraday and Kerr rotation with strained graphene,” Opt. Lett. 37, 3237 (2012)
[Crossref] [PubMed]

Tanabe, S.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

Thitapura, T.

Tokura, Y.

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]

Tse, W-K.

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

Tsukazaki, A.

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]

van der Marel, D.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Viana-Gomes, J.

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

Walter, A. L.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

Wang, H-L.

S-J. Zhang, H. Pan, and H-L. Wang, “Frequency dependent optical conductivity of strained graphene at T=0 from an effective quantum field theory,” Physica B 511, 80 (2017).
[Crossref]

Yoo, J. Y.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Yumoto, G.

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

Zhang, H.

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

Zhang, S. C.

B. A. Bernevig, T. L. Hughes, and S. C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757 (2006).
[Crossref] [PubMed]

Zhang, S-C.

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

Zhang, S-J.

S-J. Zhang, H. Pan, and H-L. Wang, “Frequency dependent optical conductivity of strained graphene at T=0 from an effective quantum field theory,” Physica B 511, 80 (2017).
[Crossref]

J. Phys. B: At. Mol. Opt. Phys. (1)

P. Srisangyingcharoen, R. Klinkla, and S. Boonchui, “Dynamics of propagating surface plasmon induced photon emission from quantum dots: quantum history approach,” J. Phys. B: At. Mol. Opt. Phys. 48, 215501 (2015).
[Crossref]

J. Phys. Chem. Solids. (1)

R. O. Dillon, I. L. Spain, and J. W. McClure, “Electronic energy band parameters of graphite and their dependence on pressure, temperature and acceptor concentration,” J. Phys. Chem. Solids. 38, 635 (1997).
[Crossref]

J. Phys.: Condens. Matter (2)

H. P. Paudel and M. N. Leuenberger, “Giant Faraday effect due to Pauli exclusion principle in 3D topological insulators,” J. Phys.: Condens. Matter 26, 082201 (2014).

A. J. Chaves, T. Frederico, O. Oliveira, W. de Paula, and M. C. Santos, “Optical conductivity of curved graphene,” J. Phys.: Condens. Matter 26, 185301 (2014).

Nat. Commun. (2)

R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, “Quantum Faraday and Kerr rotations in graphene,” Nat. Commun. 4, 1841 (2013).
[Crossref] [PubMed]

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]

Nat. Phys. (2)

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48 (2010).
[Crossref]

D. E. Chang, A. S. Sorensen, A. E. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807 (2007).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phy. Rev. A (1)

D. Georgiev and E. Cohen, “Probing finite coarse-grained virtual Feynman histories with sequential weak values,” Phy. Rev. A 97, 052102 (2018).
[Crossref]

Phy. Rev. B (1)

T. Danz, A. Neff, H. J. Gaida, R. Bormann, C. Ropers, and S. Schäfer, “Ultrafast sublattice pseudospin relaxation in graphene probed by polarization-resolved photoluminescence,” Phy. Rev. B 95, 241412 (2017).
[Crossref]

Phys. Rev. B (4)

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

A. L. Kitt, V. M. Pereira, A. K. Swan, and B. B. Goldberg, “Lattice-corrected strain-induced vector potentials in graphene,” Phys. Rev. B 85, 159909 (2012).
[Crossref]

V. M. Pereira and A. H. C. Neto, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80, 045401 (2009).
[Crossref]

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

Phys. Rev. lett. (3)

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

M. Laakso and M. Pletyukhov, “Scattering of two photons from two distant qubits: exact solution,” Phys. Rev. lett. 113, 183601 (2014).
[Crossref] [PubMed]

P. Dietl, F. Piechon, and G. Montambaux, “New magnetic field dependence of landau levels in a graphenelike structure,” Phys. Rev. lett. 100, 235405 (2008).
[Crossref]

Physica B (2)

S-J. Zhang, H. Pan, and H-L. Wang, “Frequency dependent optical conductivity of strained graphene at T=0 from an effective quantum field theory,” Physica B 511, 80 (2017).
[Crossref]

P. Fanbanrai, A. Hutema, and S. Boonchui, “Effects of strain on the Schwinger pair creation in graphene,” Physica B 472, 84 (2015).
[Crossref]

Rev. Mod. Phys. (2)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

N. M. R. Peres, “The transport properties of graphene: An introduction,” Rev. Mod. Phys. 82, 2673 (2010).
[Crossref]

Sci. Rep. (1)

Z. B. Siu, S. G. Tan, and M. B. A. Jalil, “Effective Hamiltonian for surface states of topological insulator nanotubes,” Sci. Rep. 7, 45350 (2017).
[Crossref] [PubMed]

Science (1)

B. A. Bernevig, T. L. Hughes, and S. C. Zhang, “Quantum spin hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757 (2006).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Show a single photon propagating in the z direction with polarization in the x direction interacts with an electron in the material. The polarization of a reflected photon and a transmitted photon is rotated in Faraday effect.
Fig. 2
Fig. 2 Reflectance R and Kerr rotation θK for single electron interacting with single photon by varying q ≡ (qx, qy) in non-strained graphene. A red line for electron with wave vector (1,7.53) ×108 m−1, blackline for (5,5.72) ×108 m−1, blue line for (1,7.53) ×107 m−1 and green line for (5,5.72) ×107 m−1 around the K valley (τ = 1). Insets in Fig. 2(b) show the plots of imaginary part which are corresponding to the critical conditions.
Fig. 3
Fig. 3 Transmittance T and Faraday rotation θF for single electron interacting with single photon by varying q ≡ (qx, qy) in non-strained graphene. A red line for electron with wave vector (1,7.53) ×108 m−1, black line (5,5.72) ×108 m−1, blue line (1,7.53) ×107 m−1 and green line (5,5.72) ×107 m−1 around the K valley (τ = 1). Insets in Fig. 3(b) show the plots of imaginary part which are corresponding to the critical conditions.
Fig. 4
Fig. 4 Show that transmittance T and reflectance R as a function of the phase of spinor for q = (5,3)×108 m−1 around the K valley (τ = 1) and Poisson’s ratio ν = 0.3 by varying strain parameter in different strain types compared with non strain case (black line). Inset in Fig. 4(b) show that the phase shift ϕ q , η ( s , τ ) , η = x , y for shear strain type as a function of stain parameter.
Fig. 5
Fig. 5 Reflectance R and Kerr rotation θK (in radian) for q = (5,3)×108 m−1 around the K valley (τ = 1) by varying strain parameter in different strain types. Using Poisson’s ratio ν = 0.3 and ε = 0 (red), ε = 0.05 (black) and ε = 0.1 (blue). Inset in Figs. 5(a), 5(c), show a little shift to the left corresponding to the energy level changing with strain.
Fig. 6
Fig. 6 Transmittance T and Faraday rotation θF (in radian) for q = (5,3)×108 m−1 around the K valley (τ = 1) by varying strain parameter in different strain types. sing Poisson’s ratio ν = 0.3 and ε = 0 (red), ε = 0.05 (black) and ε = 0.1 (blue). Inset in Figs. 6(a) and 6(c), show a little shift to the left corresponding to the energy level changing with strain.

Equations (42)

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θ F c μ 0 1 + ϵ r R e [ q σ x y ( q , ω ) ]
i t | ψ ( t ) ** = H s y s | ψ ( t ) ** .
H sys = H ti + H ph + H ph ti .
H ti = q ( c ^ c , q c ^ v , q ) H ti ( c ^ c , q c ^ v , q ) ,
H ti = H 0 ( q ) σ 0 + H 1 ( q ) σ 1 + H 2 ( q ) σ 2 ,
σ 0 = ( 1 0 0 1 ) , σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 i i 0 ) ,
ϵ λ ( q ) = H 0 ( q ) + λ ( H 1 ( q ) ) 2 + ( H 2 ( q ) ) 2 , ψ λ ( q , r ) = 1 2 ( 1 λ e i λ ϕ q ) e i k r ,
ε = ( ε x x ε x y ε x y ε y y ) .
t ( a + Δ δ i ) t 0 ( 1 ( β / a ) ( | ( 1 + ε ) δ 0 i δ 0 i | ) ) ,
δ k = π a ( 4 3 3 ε y y + β 2 π ( ε x x ε y y ) , 4 3 3 ε x y + β π ε x y ) ,
H 0 ( q ) = 0 , , H 1 ( q ) = v 1 q v F e c A p d x   , H 2 ( q ) = v 2 q v F e c A p d y
v 1 y = v F ( 1 1 4 β ( ε x x + 3 ε y y ) 2 π 3 3 ε y y ) , v 1 x = τ v F ( 1 2 β ε x y 2 π 3 3 ε x y )  
v 2 y = v F ( 1 2 β ( ε x y 2 π 3 3 ε x y ) , v 2 x = τ v F ( 1 1 4 β ( 3 ε x x + ε y y ) + 2 π 3 3 ε y y ) .
H ph = i c η = x , y d z [ b L ( η ) ( z ) z b L ( η ) ( z ) b R ( η ) ( z ) z b R ( η ) ( z ) ]
b R ( η ) ( z ) = 0 d k e i k z b k ( η ) a n d b L ( η ) ( z ) = 0 d k e i k z b k ( η ) ,
H ph ti = γ = R , L η = x , y d z ν η δ ( z ) [ b γ ( η ) ( z ) J η + b γ ( η ) ( z ) J η ]
J η = q λ = c , v λ = c , v ψ λ | j η | ψ λ c ^ λ , q c ^ λ , q .
| ψ ( q , ϵ k ) = γ = R , L η = x , y   d z E k , γ ( η ) ( z ) b γ η ( z ) | 0 , μ + e k ( η ) ( ϵ q ) P ^ c , v ( η ) ( q ) | 0 , μ ,
P ^ c , v ( η ) ( q ) = ψ c | j η | ψ v c ^ c , q c ^ v , q ,
( ϵ k ϵ μ ) E k , + ( η ) ( z ) + i c z E k , ( η ) ( z ) ν η δ ( z ) [ e k ( η ) ( q ) η , η + e k ( η ) ( q ) η , η ] = 0
η = x , y ( ϵ k η , η Λ η , η ) e k ( η ) ( q ) ν η ( E k , + ( η ) ( 0 ) η , η E k , + ( η ) ( 0 ) η , η ) = 0
η , η = ψ v | j η | ψ c ψ c | j η | ψ v n v ( q )
Λ η , η = ψ v | j η | ψ c ψ c | j η | ψ v ( ϵ μ + ϵ c ϵ v ) n v ( q ) .
2 z 2 E k , + ( η ) ( z ) + i ω μ 0 δ ( z ) η = x , y σ η , η E k , + ( η ) ( z ) + ω 2 ε r μ 0 E k , + ( η ) ( z ) = 0 ; ω k = ϵ k ϵ μ ,
σ η , η = ( i 4 π ϵ r c 2 A 0 ω k ) [ x , x x , y     y , x y , y ] [ ϵ k x , x Λ x , x ϵ k x , y Λ x , y     ϵ k y , x Λ y , x ϵ k y , y Λ y , y ] 1 [ x , x x , y     y , x y , y ] .
σ η , η = ( π ϵ r c 2 A 0 ) ψ v | j η | ψ c ψ c | j η | ψ v i ω k ( ω k ( ϵ c ϵ v ) ) n v ( q ) .
( E k , + ( x , i ) ( 0 ) , E k , + ( y , i ) ( 0 ) ) + ( E k , + ( x , r ) ( 0 ) , E k , + ( y , r ) ( 0 ) ) ) = ( E k , + ( x , t ) ( 0 ) , E k , + ( y , t ) ( 0 ) )
z E k , + ( η , t ) ( 0 ) z E k , + ( η , i ) ( 0 ) z E k , + ( η , r ) ( 0 ) = i ω μ 0 η = x , y σ η , η E k , + ( η ) ( 0 )
[ E k , + ( x , t ) ( 0 )   E k , + ( y , t ) ( 0 ) ] = [ ( 1 + Γ x x ) Γ x y     Γ y x ( 1 + Γ y y ) ] 1 [ E k , + ( x , i ) ( 0 )   E k , + ( y , i ) ( 0 ) ] ,
[ E k , + ( x , r ) ( 0 )   E k , + ( y , r ) ( 0 ) ] = [ ( 1 + Γ x x ) Γ x y     Γ y x ( 1 + Γ y y ) ] 1 [ Γ x x Γ x y     Γ y x Γ y y ] [ E k , + ( x , i ) ( 0 )   E k , + ( y , i ) ( 0 ) ] .
r ± = | r ± | e i θ K ( ± ) = | ( E k , + ( x , r ) ± i E k , + ( y , r ) ) / E k , + ( x , i ) | = ± i G Π x ( Π x ± i Π y ) 2 ω k ( ω k ( ϵ c ϵ v ) + i G ( | Π x | 2 + | Π y | 2 )
t ± = | t ± | e i θ F ( ± ) = | ( E k , + ( x , t ) ± i E k , + ( y , t ) ) / E k , + ( x , i ) | = ω k ( ω k ( ϵ c ϵ v ) ± G Π y ( Π x ± i Π y ) 2 ω k ( ω k ( ϵ c ϵ v ) + i G ( | i x | 2 + | Π y | 2 )
Π η = ± i ( v 1 η ) 2 + ( v 2 η ) 2 s i n ( ϕ q ϕ q , η ( s ) ) , ϕ q , η ( s , τ ) = t a n 1 ( v 2 η / v 1 η ) .
| t y t x | = | 2 ( ω k ( ω k ϵ c v ) ( c μ 0 / A 0 ) Π y Π y ( c μ 0 / A 0 ) Π y Π x | , ϵ c v = ϵ c ϵ v .
T ( ω ) = 1 2 ( | t + | 2 + | t | 2 ) ,    R ( ω ) = 1 2 ( | r + | 2 + | r | 2 )
θ F ( ω ) = 1 2 ( θ F ( + ) θ F ( ) ) ,    θ K ( ω ) = 1 2 ( θ K ( + ) θ K ( ) ) .
S ( 1 ) = η d k t η b R , k η | 0 , μ 0 , μ | b R , k η + d k r η b L , k η | 0 , μ 0 , μ | b L , k η .
g η , η ( 1 ) ( z 1 , z 2 ) = o u t | b R η ( z 1 ) b R η ( z 2 ) | o u t o u t | b R η ( z ) b R η ( z ) | o u t .
g η , η ( 1 ) c l ( z 1 , z 2 ) = d k E k , + ( η ) ( z 2 ) E k , + ( η ) * ( z 1 ) d k E k , + ( η ) ( z ) E k , + ( η ) * ( z ) = d k t η ( k ) t η * ( k ) e i k ( z 1 z 2 ) d k t η ( k ) t η * ( k ) .
g η , η ( 1 ) ( z 1 , z 2 ) = i n | S ( 1 ) b R η ( z 1 ) b R η ( z 2 ) S ( 1 ) | i n + n > 1 i n | S ( n ) b R η ( z 1 ) b R η ( z 2 ) S ( n ) | i n .
ω c ( ± ) = 1 2 ( ϵ c v ± ( ϵ c v ) 2 4     G   ( Π x / Π y ) (   | Π y | 2 + | Π x | 2 ) ) .
ε i s = ( ε 0 0 ε ) ,   ε s s = ( 0 ε ε 0 ) ,   ε a s = ( ε 0 0 ν ε ) ,   ε z s = ( ν ε 0 0 ε ) .

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