Abstract

We theoretically investigate the polaritonic band structure and dispersion properties of graphene using transfer matrix methods, with strongly coupled graphene plasmons (GPs) and molecular infrared vibrations as a representative example. Two common geometrical configurations are considered: graphene coupled subwavelength dielectric grating (GSWDG) and graphene nanoribbons (GNR). By exploiting the dispersion and the band structure, we show the possibility of tailoring desired polaritonic behavior in each of the two configurations. We compare the strength of coupling occurring in both structures and find that the interaction is stronger in GNR than that of GSWDG structure as a result of the stronger field confinement of the edge modes. The band structure and dispersion analysis not only sheds light on the physics of the hybridized polariton formation but also offers insight into tailoring the optical response of graphene light-matter interactions for numerous applications, such as biomolecular sensing and detection.

© 2016 Optical Society of America

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2015 (2)

A. Y. Zhu and E Cubukcu, “Graphene nanophotonic sensors,” 2D Mater. 2, 032005 (2015).
[Crossref]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. Javier García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349, 165–168 (2015).
[Crossref] [PubMed]

2014 (8)

F. Liu and E. Cubukcu, “A tunable omni-directional sensing platform: strong light-matter interactions enabled by graphene,” Proc. of SPIE 8993, 899326 (2014).

L. Wang, W. Cai, W. Luo, Z. Ma, C. Du, X. Zhang, and J. Xu, “Mid-infrared plasmon induced transparency in heterogeneous graphene ribbon pairs,” Opt. Express 22, 32450–32456 (2014).
[Crossref]

A. Y. Zhu, F. Yi, J. C. Reed, H. Zhu, and E. Cubukcu, “Optoelectromechanical multimodal biosensor with graphene active region,” Nano Lett 14, 5641–5649 (2014).
[Crossref] [PubMed]

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-BN heterostructures,” Nano Lett. 14, 3876–3880 (2014).
[Crossref] [PubMed]

I. J. Luxmoore, C. H. Gan, P. Q. Liu, F. Valmorra, P. Li, J. Faist, and G. R. Nash, “Strong Coupling in the Far-Infrared between Graphene Plasmons and the Surface Optical Phonons of Silicon Dioxide,” ACS Photon. 1, 1151–1155 (2014).
[Crossref]

Y. Li, H. Yan, D. B. Farmer, X. Meng, W. Zhu, R. M. Osgood, T. F. Heinz, and P. Avouris, “Graphene plasmon enhanced vibrational sensing of surface-adsorbed layers,” Nano Lett. 14, 1573–1577 (2014).
[Crossref] [PubMed]

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

F. Javier García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photon. 1, 135–152 (2014).
[Crossref]

2013 (8)

P. Tassin, T. Koschny, and C. M. Soukoulis, “Graphene for terahertz applications,” Science 341, 620–621 (2013).
[Crossref] [PubMed]

A. E. Schlather, N. Large, A. S. Urban, P. Nordlander, and N. J. Halas, “Near-field mediated plexcitonic coupling and giant Rabi splitting in individual metallic dimers,” Nano Lett. 13, 3281–3286 (2013).
[Crossref] [PubMed]

A. Lovera, B. Gallinet, P. Nordlander, and Olivier J.F. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano 7, 4527–4536 (2013).
[Crossref] [PubMed]

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

F. Liu and E. Cubukcu, “Tunable omnidirectional strong light-matter interactions mediated by graphene surface plasmons,” Phys. Rev. B 88, 115439 (2013).
[Crossref]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nature Photon. 7, 394–399 (2013).
[Crossref]

J. H. Strait, P. Nene, W. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined plasmons in graphene microstructures: experiments and theory,” Phys. Rev. B 87, 241410(R) (2013).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13, 2541–2547 (2013).
[Crossref] [PubMed]

2012 (8)

T. R. Zhan, F. Y. Zhao, X. H. Hu, X. H. Liu, and J. Zi, “Band structure of plasmons and optical absorption enhancement in graphene on subwavelength dielectric gratings at infrared frequencies,” Phys. Rev. B 86, 165416 (2012).
[Crossref]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano. 6, 7806–7813 (2012).
[Crossref] [PubMed]

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,” Nature Nanotech. 7, 330–334 (2012).
[Crossref]

Y. V. Bludov, N. M. R. Peres, and M. I. Vasilevskiy, “Graphene-based polaritonic crystal,” Phys. Rev. B 85, 245409 (2012).
[Crossref]

A. Salomon, R. J. Gordon, Y. Prior, T. Seideman, and M. Sukharev, “Strong coupling between molecular excited states and surface plasmon modes of a slit array in a thin metal film,” Phys. Rev. Lett. 109, 073002 (2012).
[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 nanoimaging,” Nature 487, 82–85 (2012).
[PubMed]

P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nature Photon. 6, 259–264 (2012).
[Crossref]

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

2011 (6)

T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Reversible switching of ultrastrong light-molecule coupling,” Phys. Rev. Lett. 106, 196405 (2011).
[Crossref] [PubMed]

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

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84, 161407(R) (2011).
[Crossref]

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

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nature Nanotech. 6, 630–634 (2011).
[Crossref]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene-SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[Crossref] [PubMed]

2010 (6)

Y. Liu and R. F. Willis, “Plasmon-phonon strongly coupled mode in epitaxial graphene,” Phys. Rev. B 81, 081406(R) (2010).
[Crossref]

R. J. Koch, Th. Seyller, and J. A. Schaefer, “Strong phonon-plasmon coupled modes in the graphene/silicon carbide heterosystem,” Phys. Rev. B 82, 201413(R) (2010).
[Crossref]

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dotCnanocavity system,” Nature Phys. 6, 279–283 (2010).
[Crossref]

S. Kéna-Cohen and S. R. Forrest, “Room-temperature polariton lasing in an organic single-crystal microcavity,” Nature Photon. 4, 371–375 (2010).
[Crossref]

D. Snoke and P. Littlewood, “Polariton condensates,” Phys. Today 63, 42–47 (2010).
[Crossref]

K. Aydin, I. M. Pryce, and Harry A. Atwater, “Symmetry breaking and strong coupling in planar optical metamaterials,” Opt. Express 18, 13407–13417 (2010).
[Crossref] [PubMed]

2009 (3)

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2007 (6)

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2004 (1)

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2001 (3)

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1992 (2)

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Figures (7)

Fig. 1
Fig. 1 (a) The schematic diagram of a sandwiched structure which consists of graphene layer at the interface II. The optical conductivity of graphene is denoted as σ. The dielectric constants of the sandwiched structure are ε1, ε2, and ε3, respectively. The electric or magnetic field of point A and that of point B can be related by a transmission matrix. (b) The real and imaginary parts of the dielectric constant ε2 as a function of frequency. The Drude-Lorentz model is applied to simulate the molecular dipoles in medium 2. In the calculations, ε∞ = 2, ωm = 0.111 μm−1, ωp = 2×1014 rad s−1, and γm = 1×1012 rad s−1.

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Fig. 2
Fig. 2 The analytical dispersion and the numerical absorption spectra (the colored contour) for the polaritons in GSWDG (a) and GNR (b) configuration, respectively. The red open circles in (a) and crosses in (b) are calculated by σ which describes the optical conductivity of a continuous graphene sheet. However, the red open circles in (b) are calculated with the effective conductivity σ 1 eff ( ω ). The light line is denoted as the blue dashed lines in both panels. For both structures, the band splitting at the molecular resonance are observed, which is ascribed to the anticrossing behavior of GP band and the molecular resonance (black dashed lines). It is clear that the coupling in GNR is much stronger than that in GSWDG structure.

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Fig. 3
Fig. 3 The polariton excitation configurations [(a) and (d)] and the polaritonic features under normal incidence [(b), (c), (e), and (f)]. The numerically calculated spectra in (b) and (e) show only 1.1% and 4.1% difference from the results tailored by the polariton dispersion in wavelength for GSWDG (a) and GNR (d) structure, respectively. Similarly, the bandgap can also be designed via the dispersion curves. The results show only 0.45% and 3.15% discrepancy for the structures of (a) and (d), respectively.

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Fig. 4
Fig. 4 The analytical polariton dispersion dependence on graphene doping levels for the GSWDG (a) and GNR (b) structures are simulated, respectively. The numerically calculated results for the GSWDG (c) and GNR (d) structures show excellent agreements with that of analytical results. In the calculations, Λ = 332 nm, d = 166 nm for GSWDG structure; L = 178 nm, W = 89 nm for GNR structure.

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Fig. 5
Fig. 5 (a) The polaritonic band structure (red solid line) and the light line (blue dashed line). (b) The enlarged plot of the band structure in (a) above the light line (blue dashed line). (c) The calculated absorbance as a function of in-plane wave vector and frequency. Excellent agreements between the analytical and numerical results are clearly observed. (d) The absorbance under kx = 0.05π/Λ (red solid line) and kx = 0 (blue dashed line). (e) The absorbance under the incident angle θ = 60°. Both (d) and (e) show the secondary band splitting which is ascribed to the broken symmetry. (f) and (g) Distribution of electric field intensity at the wavelength 8.66 μm in the x and z direction, respectively. All the results are obtained with the structure of GSWDG coupled to the molecular layer. The parameters of the structure are Λ = 272 nm, D = 136 nm, t = 0.3 μm, d = 1 nm, ε = 2.3, and EF = 0.4 eV.

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Fig. 6
Fig. 6 (Similar with Fig. 5 but with the structure of GNR coupled with the molecular layer. The introduction of σ 1 eff ( ω ) deduced from the fundamental plasmon modes gives rise to the polaritonic band structures in (a) and (b). Compared with Fig. 5, the band structure and the absorption spectra (c) show larger band splitting, flatter bands, and stronger absorption. Flat bands further render the insignificant secondary band splitting under oblique incidence, as shown in (d) and (e). This is ascribed by the stronger interaction between the edge mode supported by the GNR structure and the molecular dipoles. From (f) and (g) with the wavelength 8.4 μm of the incident light (marked by the black solid arrow in (d)), polaritonic standing waves occur between the neighboring ribbons, which explains the origin of the small oscillations imposed on the main absorption peak. The parameters of the structure are L = 146 nm, W = 73 nm, t = 0.3 μm, d = 1 nm, ε = 2.3, and EF = 0.4 eV.

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Fig. 7
Fig. 7 (a) The configuration of GNR structure coupled with polar substrate in one unit cell. (b) The analytical polariton dispersion from Eq. (17) (red solid circles). The GP dispersion and the substrate phonon mode are identified by the solid black lines. (c) The numerical results show excellent agreement with the prediction results by setting |q| = 35.44 μm−1 (dashed line in panel (b)). In the calculations, L = 178 nm, W = 89 nm for GNR structure, and l = 300 nm for the polar substrate.

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Equations (17)

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σ ( ω ) = 2 i e 2 k B T π ℏ 2 ( ω + i τ − 1 ) ln [ 2 cosh ( E F 2 k B T ) ] + e 2 4 ℏ [ ( 1 2 + 1 π arctan ( ℏ ω − 2 E F 2 k B T ) ) − i π ln | ℏ ω + 2 E F ℏ ω − 2 E F | ] ,
ℳ 1 → 2 = 1 2 ( 1 + η 21 1 − η 21 1 − η 21 1 + η 21 ) ,
ℳ 2 → 2 ( d ) = ( e − i k 2 z d 0 0 e i k 2 z d ) ,
ℳ 2 → 3 = 1 2 ( 1 + η 32 + ξ 23 1 − η 32 − ξ 23 1 − η 32 + ξ 23 1 + η 32 − ξ 23 ) ,
( 1 r ) = ℳ 1 → 2 ℳ 2 → 2 ( d ) ℳ 2 → 3 ( t 0 ) ,
( 1 + η 21 ) ( 1 + η 32 + ξ 23 ) e − i k 2 z d + ( 1 − η 21 ) ( 1 − η 32 + ξ 23 ) e i k 2 z d = 0 .
tanh ( k 2 z ′ d ) = − 1 + ξ 23 + η 21 η 32 ( 1 + ξ 23 ) η 21 + η 32 .
ε 2 = ε ∞ + ω p 2 ω m 2 − ω 2 − i γ m ω ,
| q | = | k x | + j | G x | ,
| q | = | G x | = 2 π / Λ .
| q | W + 2 ϕ m = m π ,         ( m = 1 , 2 , ⋯ ) ,
ω p g = σ ( ω = 0 ) | q | 2 ε 0 ε avg τ ,
ω p 0 g = α 1.156 σ ( ω = 0 ) ε 0 ε avg τ W ,
| q | W ˜ m = m π ,     ( m = 1 , 2 , ⋯ ) ,
| q | = | G x | = π / ( 1.505 W ) .
( 1 r ) = ℳ 3 → 2 ℳ 2 → 2 ( d ) ℳ 2 → 1 ( t 0 ) ,
tan ( k 2 z ′ d ) = − 1 + η 12 ( η 23 + ξ 32 ) η 12 + η 23 + ξ 32 .

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