Abstract

A two-dimensional graphene plasmonic crystal composed of periodically arranged graphene nanodisks is proposed. We show that the band topology effect due to inversion symmetry broken in the proposed plasmonic crystals is obtained by tuning the chemical potential of graphene nanodisks. Utilizing this kind of plasmonic crystal, we constructed N-shaped channels and realized topologically edged transmission within the band gap. Furthermore, topologically protected exterior boundary propagation, which is immune to backscattering, was also achieved by modifying the chemical potential of graphene nanodisks. The proposed graphene plasmonic crystals with ultracompact size are subject only to intrinsic material loss, which may find potential applications in the fields of topological plasmonics and high density nanophotonic integrated systems.

© 2017 Optical Society of America

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References

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2017 (2)

G. An, S. Li, X. Yan, X. Zhang, Z. Yuan, H. Wang, Y. Zhang, X. Hao, Y. Shao, and Z. Han, “Extra-broad photonic crystal fiber refractive index sensor based on surface plasmon resonance,” Plasmonics 12(2), 465–471 (2017).
[Crossref]

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118(24), 245301 (2017).
[Crossref] [PubMed]

2016 (5)

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref] [PubMed]

T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
[Crossref]

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U.S.A. 113(18), 4924–4928 (2016).
[Crossref] [PubMed]

J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2016).
[Crossref] [PubMed]

2015 (4)

T. Ma, A. B. Khanikaev, S. H. Mousavi, and G. Shvets, “Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides,” Phys. Rev. Lett. 114(12), 127401 (2015).
[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]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M. Janssen, M. M. Fogler, and D. N. Basov, “Edge and surface plasmons in graphene nanoribbons,” Nano Lett. 15(12), 8271–8276 (2015).
[Crossref] [PubMed]

2014 (4)

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref] [PubMed]

W. Qiu, X. Liu, J. Zhao, S. He, Y. Ma, J.-X. Wang, and J. Pan, “Nanofocusing of mid-infrared electromagnetic waves on graphene monolayer,” Appl. Phys. Lett. 104(4), 041109 (2014).
[Crossref]

J. N. Dash and R. Jha, “Graphene-based birefringent photonic crystal fiber sensor using surface plasmon resonance,” IEEE Photonics Technol. Lett. 26(11), 1092–1095 (2014).
[Crossref]

J. Zhao, X. Liu, W. Qiu, Y. Ma, Y. Huang, J.-X. Wang, K. Qiang, and J.-Q. Pan, “Surface-plasmon-polariton whispering-gallery mode analysis of the graphene monolayer coated InGaAs nanowire cavity,” Opt. Express 22(5), 5754–5761 (2014).
[Crossref] [PubMed]

2013 (3)

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

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111(24), 243905 (2013).
[Crossref] [PubMed]

Y. Hadad and B. Z. Steinberg, “Quasistatic resonance of a chemical potential interruption in a graphene layer and its polarizability: The mixed-polarity semilocalized plasmon,” Phys. Rev. B 88(7), 075439 (2013).
[Crossref]

2012 (2)

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

2011 (2)

F. H. 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]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

2010 (2)

P. Avouris, “Graphene: electronic and photonic properties and devices,” Nano Lett. 10(11), 4285–4294 (2010).
[Crossref] [PubMed]

D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010).
[Crossref] [PubMed]

2009 (3)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

X. Ao, Z. Lin, and C. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80(3), 033105 (2009).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

2008 (2)

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

2004 (1)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

2001 (1)

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

An, G.

G. An, S. Li, X. Yan, X. Zhang, Z. Yuan, H. Wang, Y. Zhang, X. Hao, Y. Shao, and Z. Han, “Extra-broad photonic crystal fiber refractive index sensor based on surface plasmon resonance,” Plasmonics 12(2), 465–471 (2017).
[Crossref]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Ao, X.

X. Ao, Z. Lin, and C. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80(3), 033105 (2009).
[Crossref]

Avouris, P.

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref] [PubMed]

P. Avouris, “Graphene: electronic and photonic properties and devices,” Nano Lett. 10(11), 4285–4294 (2010).
[Crossref] [PubMed]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Barik, S.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Basov, D. N.

Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M. Janssen, M. M. Fogler, and D. N. Basov, “Edge and surface plasmons in graphene nanoribbons,” Nano Lett. 15(12), 8271–8276 (2015).
[Crossref] [PubMed]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

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]

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

Camara, N.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Castro Neto, A. H.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Centeno, A.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Chan, C.

X. Ao, Z. Lin, and C. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80(3), 033105 (2009).
[Crossref]

Chang, D. E.

F. H. 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, J.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Chen, X.-D.

J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2016).
[Crossref] [PubMed]

Chen, Y.

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U.S.A. 113(18), 4924–4928 (2016).
[Crossref] [PubMed]

Chen, Y.-F.

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U.S.A. 113(18), 4924–4928 (2016).
[Crossref] [PubMed]

Cheng, X.

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15(5), 542–548 (2016).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Chow, E.

Christensen, T.

D. Jin, T. Christensen, M. Soljačić, N. X. Fang, L. Lu, and X. Zhang, “Infrared topological plasmons in graphene,” Phys. Rev. Lett. 118(24), 245301 (2017).
[Crossref] [PubMed]

Dai, S.

Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M. Janssen, M. M. Fogler, and D. N. Basov, “Edge and surface plasmons in graphene nanoribbons,” Nano Lett. 15(12), 8271–8276 (2015).
[Crossref] [PubMed]

Dash, J. N.

J. N. Dash and R. Jha, “Graphene-based birefringent photonic crystal fiber sensor using surface plasmon resonance,” IEEE Photonics Technol. Lett. 26(11), 1092–1095 (2014).
[Crossref]

DeGottardi, W.

S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Dominguez, G.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Dong, J.-W.

J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2016).
[Crossref] [PubMed]

Ebbesen, T. W.

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W. Qiu, X. Liu, J. Zhao, S. He, Y. Ma, J.-X. Wang, and J. Pan, “Nanofocusing of mid-infrared electromagnetic waves on graphene monolayer,” Appl. Phys. Lett. 104(4), 041109 (2014).
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L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
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G. An, S. Li, X. Yan, X. Zhang, Z. Yuan, H. Wang, Y. Zhang, X. Hao, Y. Shao, and Z. Han, “Extra-broad photonic crystal fiber refractive index sensor based on surface plasmon resonance,” Plasmonics 12(2), 465–471 (2017).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Zhang, Y.

G. An, S. Li, X. Yan, X. Zhang, Z. Yuan, H. Wang, Y. Zhang, X. Hao, Y. Shao, and Z. Han, “Extra-broad photonic crystal fiber refractive index sensor based on surface plasmon resonance,” Plasmonics 12(2), 465–471 (2017).
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W. Qiu, X. Liu, J. Zhao, S. He, Y. Ma, J.-X. Wang, and J. Pan, “Nanofocusing of mid-infrared electromagnetic waves on graphene monolayer,” Appl. Phys. Lett. 104(4), 041109 (2014).
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Zhao, Z.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Zhu, H.

J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2016).
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Zhu, S.

Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M. Janssen, M. M. Fogler, and D. N. Basov, “Edge and surface plasmons in graphene nanoribbons,” Nano Lett. 15(12), 8271–8276 (2015).
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ACS Nano (1)

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
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IEEE Photonics Technol. Lett. (1)

J. N. Dash and R. Jha, “Graphene-based birefringent photonic crystal fiber sensor using surface plasmon resonance,” IEEE Photonics Technol. Lett. 26(11), 1092–1095 (2014).
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Nat. Mater. (2)

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J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16(3), 298–302 (2016).
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Nat. Photonics (1)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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T. Ma and G. Shvets, “All-Si valley-Hall photonic topological insulator,” New J. Phys. 18(2), 025012 (2016).
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Plasmonics (1)

G. An, S. Li, X. Yan, X. Zhang, Z. Yuan, H. Wang, Y. Zhang, X. Hao, Y. Shao, and Z. Han, “Extra-broad photonic crystal fiber refractive index sensor based on surface plasmon resonance,” Plasmonics 12(2), 465–471 (2017).
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Proc. Natl. Acad. Sci. U.S.A. (1)

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U.S.A. 113(18), 4924–4928 (2016).
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Science (1)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
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Figures (5)

Fig. 1
Fig. 1 (a) Three-dimensional (3D) view of the graphene plasmonic crystals. (b) Schematic structure of the graphene plasmonic crystals, the upper and lower rights display the honeycomb unit cell and k path for the unit cell.
Fig. 2
Fig. 2 The band structures of the graphene plasmonic crystals. (a) μc2 = 0.6 eV, μc1 = μc3 = 0.3 eV. (b) μc2 = 0.6 eV, μc1 = 0.33 and μc3 = 0.27 eV. The insets are the eigen-field Ez distributions of the plasmonic crystals at K point. The radii of the graphene nanodisks and the lattice constant are set as r = 0.21a and a = 40 nm constantly in this paper.
Fig. 3
Fig. 3 (a) Schematic of zigzag interfaces formed by two inversion symmetry broken graphene plasmonic crystals and the corresponding electric field intensity distribution, μc2 = 0.6 eV, μc1 = 0.33 and μc3 = 0.27 eV. (b) Dispersion relation of edge modes calculated for a super-cell of finite period (N = 19), the grey shaded regions represent the projected band diagrams and the inset is the representative eigen-field intensity distribution of the edge modes. (c)-(e) The electric field intensity distributions for a plane wave with frequency of 48, 49 and 50 THz propagating through N-shape channels respectively.
Fig. 4
Fig. 4 (a) Schematic diagrams of multidirectional emission and localization mode constructed by graphene plasmonic crystals, μc2 = 0.6 eV, μc1 = 0.33 and μc3 = 0.27 eV. (b)-(c) The corresponding electric field intensity distributions excited by a point source with frequency of 49.65 THz.
Fig. 5
Fig. 5 (a) Schematic structure of graphene plasmonic crystal supporting edge mode on exterior boundary and the corresponding electric field intensity distribution. (b) Schematic structure of graphene plasmonic crystal supporting edge modes along two boundaries with a sharp bend and the corresponding electric field intensity distribution. μc2 = 0.6 eV, μc1 = 0.33 and μc3 = 0.27 eV.

Equations (5)

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

ε Air β 2 k 0 2 ε A i r + ε S i O 2 β 2 k 0 2 ε S i O 2 = σ g i ω ε 0 ,
β = ε 0 ε A i r + ε S i O 2 2 2 i ω σ g .
σ g = σ i n t r a + σ i n t e r
σ i n t r a = i e 2 k B T π 2 ( ω + i / τ ) [ μ c k B T + 2 ln ( 1 + exp ( μ c k B T ) ) ] ,
σ i n t e r = i e 2 4 π ln [ 2 | μ c | ( ω + i / τ ) 2 | μ c | + ( ω + i / τ ) ] ,

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