Abstract

We show that successions of regions with different electrostatic potentials in graphene traversed by obliquely incident electrons can be designed to have the same transmission probability as the transmittance of optical layered structures illuminated with normally incident TE waves. This quantitative analogy holds, although in some cases in a limited range of parameters, despite the difference between the Dirac equation satisfied by the quantum electron wavefunction in graphene and the Helmholtz equation obeyed by the electromagnetic field.

© 2011 Optical Society of America

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  1. 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, 109–162 (2009).
    [CrossRef]
  2. Y. H. Wu, T. Yu, and Z. X. Shen, “Two-dimensional carbon nanostructures: fundamental properties, synthesis, characterization, and potential applications,” J. Appl. Phys. 108, 071301 (2010).
    [CrossRef]
  3. G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
    [CrossRef]
  4. D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188(1999).
    [CrossRef]
  5. D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2002).
  6. J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p–n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
    [CrossRef]
  7. P. Darancet, V. Olevano, and D. Mayou, “Coherent electronic transport through graphene constriction: subwavelength regime and optical analogy,” Phys. Rev. Lett. 102, 136803 (2009).
    [CrossRef] [PubMed]
  8. C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
    [CrossRef] [PubMed]
  9. M. Sharma and S. Ghosh, “Electron transport and Goos–Hänchen shift in graphene with electric and magnetic barriers: optical analogy and band structure,” J. Phys. Condens. Matter 23, 055501 (2011).
    [CrossRef] [PubMed]
  10. V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p–n junctions,” Science 315, 1252–1255 (2007).
    [CrossRef] [PubMed]
  11. A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry-Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
    [CrossRef] [PubMed]
  12. S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
    [CrossRef] [PubMed]
  13. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
    [CrossRef]
  14. S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104, 043903/1–4 (2010).
    [CrossRef]
  15. D. Dragoman, “Polarization optics analogy of quantum wavefunctions in graphene,” J. Opt. Soc. Am. B 27, 1325–1331 (2010).
    [CrossRef]
  16. D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
    [CrossRef]
  17. M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
    [CrossRef]
  18. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  19. H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
    [CrossRef] [PubMed]

2011

M. Sharma and S. Ghosh, “Electron transport and Goos–Hänchen shift in graphene with electric and magnetic barriers: optical analogy and band structure,” J. Phys. Condens. Matter 23, 055501 (2011).
[CrossRef] [PubMed]

2010

Y. H. Wu, T. Yu, and Z. X. Shen, “Two-dimensional carbon nanostructures: fundamental properties, synthesis, characterization, and potential applications,” J. Appl. Phys. 108, 071301 (2010).
[CrossRef]

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104, 043903/1–4 (2010).
[CrossRef]

H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
[CrossRef] [PubMed]

D. Dragoman, “Polarization optics analogy of quantum wavefunctions in graphene,” J. Opt. Soc. Am. B 27, 1325–1331 (2010).
[CrossRef]

2009

S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
[CrossRef] [PubMed]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[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, 109–162 (2009).
[CrossRef]

P. Darancet, V. Olevano, and D. Mayou, “Coherent electronic transport through graphene constriction: subwavelength regime and optical analogy,” Phys. Rev. Lett. 102, 136803 (2009).
[CrossRef] [PubMed]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

2008

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry-Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

2007

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p–n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p–n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

2006

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

1999

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188(1999).
[CrossRef]

1991

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

Akhmerov, A. R.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Altshuler, B. L.

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p–n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Avouris, P.

H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
[CrossRef] [PubMed]

Beenakker, C. W. J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

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, 109–162 (2009).
[CrossRef]

Cheianov, V. V.

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p–n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Chiu, H.-Y.

H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
[CrossRef] [PubMed]

Cserti, J.

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p–n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

Darancet, P.

P. Darancet, V. Olevano, and D. Mayou, “Coherent electronic transport through graphene constriction: subwavelength regime and optical analogy,” Phys. Rev. Lett. 102, 136803 (2009).
[CrossRef] [PubMed]

de Dood, M. J. A.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104, 043903/1–4 (2010).
[CrossRef]

Dragoman, D.

D. Dragoman, “Polarization optics analogy of quantum wavefunctions in graphene,” J. Opt. Soc. Am. B 27, 1325–1331 (2010).
[CrossRef]

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188(1999).
[CrossRef]

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2002).

Dragoman, M.

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188(1999).
[CrossRef]

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2002).

Fal’ko, V.

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p–n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Gaylord, T. K.

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[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, 109–162 (2009).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

Ghosh, S.

M. Sharma and S. Ghosh, “Electron transport and Goos–Hänchen shift in graphene with electric and magnetic barriers: optical analogy and band structure,” J. Phys. Condens. Matter 23, 055501 (2011).
[CrossRef] [PubMed]

S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
[CrossRef] [PubMed]

Glytsis, E. N.

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[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, 109–162 (2009).
[CrossRef]

Henderson, G. N.

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

Katsnelson, M. I.

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

Levitov, L. S.

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry-Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

Lin, Y.-M.

H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
[CrossRef] [PubMed]

Mayou, D.

P. Darancet, V. Olevano, and D. Mayou, “Coherent electronic transport through graphene constriction: subwavelength regime and optical analogy,” Phys. Rev. Lett. 102, 136803 (2009).
[CrossRef] [PubMed]

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, 109–162 (2009).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[CrossRef]

Olevano, V.

P. Darancet, V. Olevano, and D. Mayou, “Coherent electronic transport through graphene constriction: subwavelength regime and optical analogy,” Phys. Rev. Lett. 102, 136803 (2009).
[CrossRef] [PubMed]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[CrossRef]

Pályi, A.

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p–n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

Perebeinos, V.

H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
[CrossRef] [PubMed]

Peres, N. M. R.

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, 109–162 (2009).
[CrossRef]

Péterfalvi, C.

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p–n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

Rudner, M. S.

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry-Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

Sepkhanov, R. A.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Sharma, M.

M. Sharma and S. Ghosh, “Electron transport and Goos–Hänchen shift in graphene with electric and magnetic barriers: optical analogy and band structure,” J. Phys. Condens. Matter 23, 055501 (2011).
[CrossRef] [PubMed]

S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
[CrossRef] [PubMed]

Shen, Z. X.

Y. H. Wu, T. Yu, and Z. X. Shen, “Two-dimensional carbon nanostructures: fundamental properties, synthesis, characterization, and potential applications,” J. Appl. Phys. 108, 071301 (2010).
[CrossRef]

Shytov, A. V.

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry-Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

Tworzydlo, J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Wu, Y. H.

Y. H. Wu, T. Yu, and Z. X. Shen, “Two-dimensional carbon nanostructures: fundamental properties, synthesis, characterization, and potential applications,” J. Appl. Phys. 108, 071301 (2010).
[CrossRef]

Yu, T.

Y. H. Wu, T. Yu, and Z. X. Shen, “Two-dimensional carbon nanostructures: fundamental properties, synthesis, characterization, and potential applications,” J. Appl. Phys. 108, 071301 (2010).
[CrossRef]

Zandbergen, S. R.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104, 043903/1–4 (2010).
[CrossRef]

Appl. Phys. Lett.

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

J. Appl. Phys.

Y. H. Wu, T. Yu, and Z. X. Shen, “Two-dimensional carbon nanostructures: fundamental properties, synthesis, characterization, and potential applications,” J. Appl. Phys. 108, 071301 (2010).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
[CrossRef] [PubMed]

M. Sharma and S. Ghosh, “Electron transport and Goos–Hänchen shift in graphene with electric and magnetic barriers: optical analogy and band structure,” J. Phys. Condens. Matter 23, 055501 (2011).
[CrossRef] [PubMed]

Nano Lett.

H.-Y. Chiu, V. Perebeinos, Y.-M. Lin, and P. Avouris, “Controllable p–n junction formation in monolayer graphene using electrostatic substrate engineering,” Nano Lett. 10, 4634–4639 (2010).
[CrossRef] [PubMed]

Nat. Phys.

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

Phys. Rev. B

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[CrossRef]

Phys. Rev. Lett.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104, 043903/1–4 (2010).
[CrossRef]

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry-Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p–n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

P. Darancet, V. Olevano, and D. Mayou, “Coherent electronic transport through graphene constriction: subwavelength regime and optical analogy,” Phys. Rev. Lett. 102, 136803 (2009).
[CrossRef] [PubMed]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydlo, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Proc. IEEE

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

Prog. Quantum Electron.

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188(1999).
[CrossRef]

Rev. Mod. Phys.

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, 109–162 (2009).
[CrossRef]

Science

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p–n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Other

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2002).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

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Figures (4)

Fig. 1
Fig. 1

Quantum electron wavefunction in graphene propagating from an incident region with V in = 0 into an outgoing region with V out 0 (a) put into correspondence to TE waves normally incident from air, with n in = 1 , (b) into a region with refractive index n opt 1 .

Fig. 2
Fig. 2

Dependence of the incident angle of electrons in graphene on (a) electrostatic potential and (b) Fermi wavenumber, for which the transmission is the same as for TE waves at an interface between air and a medium with refractive index equal to 1.5.

Fig. 3
Fig. 3

Dependence of the width of a gated region in graphene on (a) electrostatic potential and (b) Fermi wavenumber, for which the transmission is the same as for TE waves with λ 0 = 1 μm propagating through a slab with n = 1.5 , D = 10 μm surrounded by air.

Fig. 4
Fig. 4

Dependence of the spacing between periodic gated regions in graphene on (a) electrostatic potential and (b) Fermi wavenumber, for which the transmission is the same as for TE waves with λ 0 = 1 μm propagating through slabs with n = 1.5 , D 1 = 10 μm separated by air regions of width D 2 = 15 μm .

Equations (17)

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

v F ( 0 k x i k y k x + i k y 0 ) ( ψ 1 ψ 2 ) = ( E V ) ( ψ 1 ψ 2 ) ,
( ψ 1 ψ 2 ) in = 1 cos φ ( cos ( k x L + φ ) ( i / s ) sin ( k x L ) i s sin ( k x L ) cos ( k x L φ ) ) ( ψ 1 ψ 2 ) out = M ( ψ 1 ψ 2 ) out ,
( ψ 1 ψ 2 ) in = ( 1 + r gr s in [ exp ( i φ in ) r gr exp ( i φ in ) ] ) = M tot ( ψ 1 ψ 2 ) out = ( m 11 m 12 m 21 m 22 ) ( t gr s out t gr exp ( i φ out ) ) ,
r gr = s in exp ( i φ in ) [ m 11 + s out m 12 exp ( i φ out ) ] [ m 21 + s out m 22 exp ( i φ out ) ] s in exp ( i φ in ) [ m 11 + s out m 12 exp ( i φ out ) ] + [ m 21 + s out m 22 exp ( i φ out ) ] ,
t gr = 2 s in cos φ in s in exp ( i φ in ) [ m 11 + s out m 12 exp ( i φ out ) ] + [ m 21 + s out m 22 exp ( i φ out ) ] ,
( e z h y ) in = ( cos ( q x D ) ( i / p ) sin ( q x D ) i p sin ( q x D ) cos ( q x D ) ) ( e z h y ) out = G ( e z h y ) out .
( e z h y ) in = ( 1 + r opt p in ( 1 r opt ) ) = G tot ( e z h y ) out = ( g 11 g 12 g 21 g 22 ) ( t opt p out t opt ) .
r opt = p in ( g 11 p out g 12 ) + ( g 21 p out g 22 ) p in ( g 11 p out g 12 ) ( g 21 p out g 22 ) ,
t opt = 2 p in p in ( g 11 p out g 12 ) ( g 21 p out g 22 ) ,
T gr = 4 s in s out cos φ in cos φ out | s in exp ( i φ in ) + s out exp ( i φ out ) | 2 = 2 s in s out cos φ in cos φ out 1 + s in s out cos ( φ in + φ out ) ,
T opt = 4 p in p out | p in + p out | 2 = 4 n in n out ( n in + n out ) 2 ,
s in s out sin φ in sin φ out 1 s in s out cos φ in cos φ out = 1 2 n in 2 + n out 2 n in n out .
T gr = ( cos φ in cos φ ) 2 | cos ( k x L ) cos φ in cos φ + i sin ( k x L ) ( sin φ in sin φ 1 / s ) | 2 ,
T opt = 4 n in 2 | 2 n in cos ( q x D ) i ( n in 2 / n + n ) sin ( q x D ) | 2 ,
k x L = q x D
F tot = F N = [ f 11 U N 1 ( a ) U N 2 ( a ) f 12 U N 1 ( a ) f 21 U N 1 ( a ) f 22 U N 1 ( a ) U N 2 ( a ) ] ,
k x 1 L 1 = q x 1 D 1 , k x 2 L 2 = q x 2 D 2 .

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