Abstract

Plasmons can be supported on graphene sheets as the Dirac electrons oscillate collectively. A tight-binding model for graphene plasmons is a good description as the field confinement in the normal direction is strong. With this model, the topological properties of plasmonic bands in multilayer graphene systems are investigated. The Zak phases of periodic graphene sheet arrays are obtained for different configurations. Analogous to Su-Schrieffer-Heeger (SSH) model in electronic systems, topological edge plasmon modes emerge when two periodic graphene sheet arrays with different Zak phases are connected. Interestingly, the dispersion of these topological edge modes is the same as that in the monolayer graphene and is invariant as the geometric parameters of the structure such as the separation and period change. These plasmonic edge states in multilayer graphene systems can be further tuned by electrical gating or chemical doping.

© 2015 Optical Society of America

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References

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    [Crossref]

2015 (3)

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015).
[Crossref] [PubMed]

2014 (6)

R. L. Chern, D. Han, Z. Q. Zhang, and C. T. Chan, “Additional waves in the graphene layered medium,” Opt. Express 22(26), 31677–31690 (2014).
[Crossref] [PubMed]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014).
[Crossref] [PubMed]

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

2013 (5)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38(11), 1912–1914 (2013).
[Crossref] [PubMed]

2012 (4)

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

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

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

2011 (2)

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

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

2010 (1)

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

2007 (1)

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

2006 (1)

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

2004 (1)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

1995 (1)

1989 (1)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989).
[Crossref] [PubMed]

1988 (1)

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

Avouris, P.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Belov, P. A.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

Béri, B.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Chan, C. T.

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015).
[Crossref] [PubMed]

R. L. Chern, D. Han, Z. Q. Zhang, and C. T. Chan, “Additional waves in the graphene layered medium,” Opt. Express 22(26), 31677–31690 (2014).
[Crossref] [PubMed]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Chandra, B.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Chang, D. E.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Chen, H.

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

Chen, W. J.

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, X. D.

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chern, R. L.

Dai, Y.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Das Sarma, S.

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

Dong, J. W.

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Fan, Y.

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

Fang, F.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

Freitag, M.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Fung, K. H.

Gao, W.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

García de Abajo, F. J.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

García-Vidal, F. J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Grigorenko, A. N.

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

Guinea, F.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

Hafezi, M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Han, D.

Hasan, M. Z.

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

Heeger, A. J.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

Huang, X. Q.

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Hwang, E. H.

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

Iorsh, I. V.

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

Ivchenko, E. L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014).
[Crossref] [PubMed]

Jiang, S. J.

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Joannopoulos, J. D.

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

Kane, C. L.

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

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Kivelson, S.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

Kivshar, Y. S.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

Koppens, F. H. L.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Lawrence, M.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Li, H.

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

Li, J.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Li, X.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Ling, C. W.

Liu, F.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Liu, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Lu, L.

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

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Markel, V. A.

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Miroshnichenko, A. E.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Mukhin, I. S.

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

Novoselov, K. S.

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

Pilozzi, L.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014).
[Crossref] [PubMed]

Poddubny, A. N.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014).
[Crossref] [PubMed]

Polini, M.

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

Poshakinskiy, A. V.

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014).
[Crossref] [PubMed]

Qi, X. L.

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

Schomerus, H.

Schrieffer, J. R.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

Shadrivov, I. V.

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

Shi, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Slobozhanyuk, A. P.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

Soljacic, M.

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

Sols, F.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

Soukoulis, C. M.

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

Stauber, T.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

Su, W.-P.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

Taylor, J. M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Teng, J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Tse, W. K.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Tulevski, G.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Wang, B.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

Wei, Z.

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

Wu, Y.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Wunsch, B.

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

Xia, F.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Xiao, M.

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015).
[Crossref] [PubMed]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Yan, H.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Yang, B.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Yu, S. F.

Yuan, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Zak, J.

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989).
[Crossref] [PubMed]

Zhan, T.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Zhang, S.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Zhang, S. C.

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

Zhang, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

Zhang, Z. Q.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

R. L. Chern, D. Han, Z. Q. Zhang, and C. T. Chan, “Additional waves in the graphene layered medium,” Opt. Express 22(26), 31677–31690 (2014).
[Crossref] [PubMed]

Zhou, L.

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Zhu, B.

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Zhu, W.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Zi, J.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

J. Phys. Condens. Matter (1)

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Nano Lett. (1)

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Nat. Commun. (1)

W. J. Chen, S. J. Jiang, X. D. Chen, B. Zhu, L. Zhou, J. W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Nat. Mater. (1)

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Nat. Photonics (3)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

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

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

New J. Phys. (1)

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (5)

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Photonic band gap of a graphene-embedded quarter-wave stack,” Phys. Rev. B 88(24), 241403 (2013).
[Crossref]

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75(20), 205418 (2007).
[Crossref]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

Phys. Rev. Lett. (5)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

A. V. Poshakinskiy, A. N. Poddubny, L. Pilozzi, and E. L. Ivchenko, “Radiative topological states in resonant photonic crystals,” Phys. Rev. Lett. 112(10), 107403 (2014).
[Crossref] [PubMed]

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic Metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Phys. Rev. X (1)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Rev. Mod. Phys. (3)

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]

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60(3), 781–850 (1988).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Dispersions of graphene plasmons. (a) for monolayer graphene, and (b) for triple layer sheets. In (a), the dielectric constants of background media are taken to be ε 1 = 1, ε 2 = 2. In (b), the background dielectric constant and separation distance of two adjacent graphene sheets are set to be ε = 1, d = 0.1, respectively. The black and blue curves are calculated by the tight-binding method. Only the nearest-neighbor interactions are considered for the blue dashed curves, while the next-nearest neighbor interactions are also included for the black solid curves. The red dotted curves are the rigorous result calculated using the transfer matrix method. Here Ω = ħω/E F, ky = ħcky ,SI/E F and d = d SI E F/(ħc) are the dimensionless frequency, the parallel component of the wave vector, and the separation distance, respectively. E F is the chemical potential, ky ,SI and d SI are the corresponding wave vector and separation distance in the SI units.
Fig. 2
Fig. 2 (a) Schematic view of a periodic graphene sheet array. In each unit cell, there are two graphene sheets, denoted by A and B respectively. The separation distance of the two adjacent graphene sheets within a unit cell and the period are denoted respectively by t and d. (b) Band structures with ky = 30, d = 0.2 fixed. The black, blue and red curves represent t = 0.5d, t = 0.4d and t = 0.3d, respectively. The solid and dashed curves are calculated respectively by the tight-binding method and transfer matrix method. (c) Projected band structures as t = 0.4d and d = 0.2. The red and grey parts stand for the results calculated by the tight-binding method and transfer matrix method, respectively. In (c), only the nearest- neighbor interactions are considered, while both the nearest and next-nearest neighbor interactions are taken into account in (d). We choose ε 1 = ε 2 = 1 here.
Fig. 3
Fig. 3 Edge states between “dimerized” graphene sheet arrays. (a) Schematic view of the geometric structure of connected graphene sheet arrays. The separation distances for left and right half-space are respectively t = tL and t = tR . Both the left and right half-space have the same period d. (b) The eigen-frequencies for a finite system with 61 graphene sheets, where tL = 0.6d, tR = 0.4d and d = 0.2. The eigenfunctions of surface current density Jn for three different states, one lying in the gap, the other two in the pass bands, are shown in the right panel. (c) Eigen-frequencies as a function of tR , where tL is set to be d-tR . The dielectric constants are ε 1 = ε 2 = 1 here.
Fig. 4
Fig. 4 Dispersion of topological edge modes. A finite structure (totally 13 graphene sheets) is considered with 3 periods of “dimerized” graphene sheet structure in both the left and right side. Plasmon dispersions calculated using the transfer matrix method are shown by the blue curves. The red curves represent the edge mode dispersion calculated by α(Ω) = 0 analytically. (a) Dispersions for the structure with different choices of the unit cell, tL = 0.6d and tR = 0.4d. The period and dielectric constants are respectively d = 0.2 and ε 1 = ε 2 = 1.5. (b) Dispersions for the graphene sheets embedded into the interfaces of a one-dimensional photonic crystal with dielectric constants ε 1 = 1.5 and ε 2 = 4. The separation distances tL = tR = 0.5 d have the same value here, and the period is chosen to be d = 10. The grey background stand for the projected band structures of the corresponding infinite periodic structures.

Equations (14)

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σ ˜ σ ε 0 c = 4 α i Ω + π α [ θ ( Ω 2 ) + i π ln | Ω 2 Ω + 2 | ] ,
ε 1 ε 1 k 0 2 k y 2 + ε 2 ε 2 k 0 2 k y 2 = σ ˜ k 0 ,
J n = σ m E m , y ( z n ) = k z σ ˜ 2 k 0 ε m J m e i k z | z m z n | .
α ( Ω ) J n = { J n 1 e i k n 1 , z ( d t ) + J n + 1 e i k n + 1 , z t , for n is even J n 1 e i k n 1 , z t + J n + 1 e i k n + 1 , z ( d t ) , for n is odd
J n ( q ) = { J A ( q ) e i q n d / 2 , for n is even J B ( q ) e i q ( n 1 ) d / 2 , for n is odd
[ 0 a 12 ( q ) a 21 ( q ) 0 ] [ J A J B ] = α ( Ω ) [ J A J B ]
α ( Ω ) = ± a 12 ( q ) a 21 ( q ) .
[ J A J B ] = 1 2 [ ± e i ϕ ( q ) 1 ] ,
γ = i π / d π / d ( J A * J A q + J B * J B q ) d q = ϕ ( π / d ) ϕ ( π / d ) 2 .
{ α ( Ω ) J 0 , A = J 1 , B e i k z ( d t ) + J 1 , B e i k z t + J 2 , A e i k z d + J 2 , A e i k z d α ( Ω ) J 1 , B = J 0 , A e i k z t + J 2 , A e i k z ( d t ) + J 1 , B e i k z d + J 3 , B e i k z d .
[ a 11 ( q ) a 12 ( q ) a 21 ( q ) a 22 ( q ) ] [ J A J B ] = α ( Ω ) [ J A J B ] ,
α ( Ω ) = ± a 12 ( q ) a 21 ( q ) + a 11 = ± s + f ( q ) + f ( q ) ,
a 11 ( q ) = a 22 ( q ) = 2 l = 1 ( m - 1 ) / 2 cos ( q l d ) e i k z l d , a 12 ( q ) = e i k z ( d t ) e i q d ( 1 + l = 1 ( m 1 ) / 2 e i k z l d e i q l d ) + e i k z t ( 1 + l = 1 ( m 1 ) / 2 e i k z l d e i q l d ) , a 21 ( q ) = e i k z t ( 1 + l = 1 ( m 1 ) / 2 e i k z l d e i q l d ) + e i k z ( d t ) e i q d ( 1 + l = 1 ( m 1 ) / 2 e i k z l d e i q l d ) .
a 11 ( q ) = a 22 ( q ) = 2 l = 1 m / 2 cos ( q l d ) e i k z l d , a 12 ( q ) = e i k z ( d t ) e i q d ( 1 + l = 1 m / 2 1 e i k z l d e i q l d ) + e i k z t ( 1 + l = 1 m / 2 1 e i k z l d e i q l d ) , a 21 ( q ) = e i k z t ( 1 + l = 1 m / 2 1 e i k z l d e i q l d ) + e i k z ( d t ) e i q d ( 1 + l = 1 m / 2 1 e i k z l d e i q l d ) .

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