Abstract

A simple, approximate theoretical model of surface plasmon resonance in two-dimensional metal nanoshells is developed. The model is based on the concept of short-range surface plasmons propagating around closed circular metal nanotubes. In this model, the plasmon resonance in a metal nanotube is treated as a propagating, self-interfering plasmonic wave, in a ring-type resonance, at plasmonic wavelengths matching an integer fraction of the nanotube’s effective circumference. The model is validated by detailed computer simulations based on the finite-difference time-domain method and is shown to be in full agreement with the widely used plasmon hybridization model, which is based on the quasi-static approximation.

© 2011 Optical Society of America

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  1. J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
    [CrossRef] [PubMed]
  2. A. Shalabney and I. Abdulhalim, “Sensitivity-enhancement methods for surface plasmon sensors,” Laser Photon. Rev. 5, 571–606 (2011).
    [CrossRef]
  3. G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering, K.Kneipp, M.Moskovits, and H.Kneipp, eds. (Springer-Verlag, 2006), pp. 19–46.
    [CrossRef]
  4. H. R. Stuart and D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
    [CrossRef]
  5. D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
    [CrossRef]
  6. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  7. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
    [CrossRef]
  8. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
    [CrossRef] [PubMed]
  9. T. Sondergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
    [CrossRef]
  10. G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, “Efficient suppression of radiation damping in resonant retardation-based plasmonic structures,” Phys. Rev. B 79, 113410 (2009).
    [CrossRef]
  11. E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
    [CrossRef]
  12. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  13. D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
    [CrossRef]
  14. A. Moradi, “Plasmon hybridization in metallic nanotubes,” J. Phys. Chem. Solids 69, 2936–2938 (2008).
    [CrossRef]
  15. http://www.lumerical.com/.
  16. A. Moradi, “Plasmon hybridization in tubular metallic nanostructures,” Physica B 405, 2466–2469 (2010).
    [CrossRef]
  17. O. Guilatt, B. Apter, and U. Efron, “Light absorption enhancement in thin silicon film by embedded metallic nanoshells,” Opt. Lett. 35, 1139–1141 (2010).
    [CrossRef] [PubMed]
  18. O. Guilatt, B. Apter, and U. Efron, “Light absorption enhancement in thin silicon film by embedded metallic nanoshells: erratum,” Opt. Lett. 36, 1239 (2011).
    [CrossRef]

2011 (3)

A. Shalabney and I. Abdulhalim, “Sensitivity-enhancement methods for surface plasmon sensors,” Laser Photon. Rev. 5, 571–606 (2011).
[CrossRef]

E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
[CrossRef]

O. Guilatt, B. Apter, and U. Efron, “Light absorption enhancement in thin silicon film by embedded metallic nanoshells: erratum,” Opt. Lett. 36, 1239 (2011).
[CrossRef]

2010 (2)

2009 (1)

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, “Efficient suppression of radiation damping in resonant retardation-based plasmonic structures,” Phys. Rev. B 79, 113410 (2009).
[CrossRef]

2008 (1)

A. Moradi, “Plasmon hybridization in metallic nanotubes,” J. Phys. Chem. Solids 69, 2936–2938 (2008).
[CrossRef]

2007 (1)

T. Sondergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[CrossRef]

2005 (1)

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

2003 (2)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef] [PubMed]

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef] [PubMed]

1998 (1)

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[CrossRef]

1996 (1)

H. R. Stuart and D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Abdulhalim, I.

A. Shalabney and I. Abdulhalim, “Sensitivity-enhancement methods for surface plasmon sensors,” Laser Photon. Rev. 5, 571–606 (2011).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Ahmed, W.

E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
[CrossRef]

Apter, B.

Averitt, R. D.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[CrossRef]

Bozhevolnyi, S. I.

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, “Efficient suppression of radiation damping in resonant retardation-based plasmonic structures,” Phys. Rev. B 79, 113410 (2009).
[CrossRef]

T. Sondergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[CrossRef]

Della Valle, G.

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, “Efficient suppression of radiation damping in resonant retardation-based plasmonic structures,” Phys. Rev. B 79, 113410 (2009).
[CrossRef]

Efron, U.

Feng, B.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Guilatt, O.

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef] [PubMed]

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[CrossRef]

Hall, D. G.

H. R. Stuart and D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

Homola, J.

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef] [PubMed]

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kooij, E. S.

E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
[CrossRef]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Moradi, A.

A. Moradi, “Plasmon hybridization in tubular metallic nanostructures,” Physica B 405, 2466–2469 (2010).
[CrossRef]

A. Moradi, “Plasmon hybridization in metallic nanotubes,” J. Phys. Chem. Solids 69, 2936–2938 (2008).
[CrossRef]

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef] [PubMed]

Oldenburg, S. J.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[CrossRef]

Poelsema, B.

E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
[CrossRef]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef] [PubMed]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef] [PubMed]

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

Schaadt, D. M.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Schatz, G. C.

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering, K.Kneipp, M.Moskovits, and H.Kneipp, eds. (Springer-Verlag, 2006), pp. 19–46.
[CrossRef]

Shalabney, A.

A. Shalabney and I. Abdulhalim, “Sensitivity-enhancement methods for surface plasmon sensors,” Laser Photon. Rev. 5, 571–606 (2011).
[CrossRef]

Sondergaard, T.

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, “Efficient suppression of radiation damping in resonant retardation-based plasmonic structures,” Phys. Rev. B 79, 113410 (2009).
[CrossRef]

T. Sondergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[CrossRef]

Stuart, H. R.

H. R. Stuart and D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

Van Duyne, R. P.

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering, K.Kneipp, M.Moskovits, and H.Kneipp, eds. (Springer-Verlag, 2006), pp. 19–46.
[CrossRef]

Westcott, S. L.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[CrossRef]

Young, M. A.

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering, K.Kneipp, M.Moskovits, and H.Kneipp, eds. (Springer-Verlag, 2006), pp. 19–46.
[CrossRef]

Yu, E. T.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Zandvliet, H. J. W.

E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
[CrossRef]

Anal. Bioanal. Chem. (1)

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

H. R. Stuart and D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86, 063106 (2005).
[CrossRef]

Chem. Phys. Lett. (1)

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243–247 (1998).
[CrossRef]

J. Appl. Phys. (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Phys. Chem. C (1)

E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, “Localized plasmons in noble metal nanospheroids,” J. Phys. Chem. C 115, 10321–10332 (2011).
[CrossRef]

J. Phys. Chem. Solids (1)

A. Moradi, “Plasmon hybridization in metallic nanotubes,” J. Phys. Chem. Solids 69, 2936–2938 (2008).
[CrossRef]

Laser Photon. Rev. (1)

A. Shalabney and I. Abdulhalim, “Sensitivity-enhancement methods for surface plasmon sensors,” Laser Photon. Rev. 5, 571–606 (2011).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (2)

T. Sondergaard and S. I. Bozhevolnyi, “Slow-plasmon resonant nanostructures: scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[CrossRef]

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, “Efficient suppression of radiation damping in resonant retardation-based plasmonic structures,” Phys. Rev. B 79, 113410 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

Physica B (1)

A. Moradi, “Plasmon hybridization in tubular metallic nanostructures,” Physica B 405, 2466–2469 (2010).
[CrossRef]

Science (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef] [PubMed]

Other (3)

G. C. Schatz, M. A. Young, and R. P. Van Duyne, “Electromagnetic mechanism of SERS,” in Surface-Enhanced Raman Scattering, K.Kneipp, M.Moskovits, and H.Kneipp, eds. (Springer-Verlag, 2006), pp. 19–46.
[CrossRef]

http://www.lumerical.com/.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Supplementary Material (2)

» Media 1: MPG (3710 KB)     
» Media 2: MPG (3762 KB)     

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Figures (10)

Fig. 1
Fig. 1

IMI stack. Plasmon propagation is along the x axis.

Fig. 2
Fig. 2

Formation of a closed nanoring-type structure by bend ing the IMI stack: (a) planar metal slab, (b) bent metal slab, and (c) circular metal tube. Propagating plasmon waves are shown schematically by arrows.

Fig. 3
Fig. 3

Dispersion curves for silver slab in vacuum with varying thicknesses calculated from Eq. (1) [(a) full calculated range, (b) zoomed area]. The solid curves are marked by corresponding thicknesses of the planar slab. Calculations are based on the simplified Drude model with negligible damping constant, representing the complex refractive index of the silver slab.

Fig. 4
Fig. 4

FDTD simulation setup.

Fig. 5
Fig. 5

Extinction spectrum (FDTD simulation) for a silver nanotube embedded in vacuum, with R 1 = 12 nm and R 2 = 20 nm (solid curve). Vertical dashed lines indicate positions of the first- and second-order resonances according to the NRR model.

Fig. 6
Fig. 6

Extinction spectra and tangential power flow calculated for circular and elliptical nanotubes. (a) Nanotubes with power flow monitors (radial lines); (b) extinction spectra of circular nanotube (dotted curve) and elliptical nanotubes at 0 ° , 22.5 ° and 45 ° inclination (overlapping solid curves); (c) tangential power flow along circular and 0 ° -inclined elliptical nanotubes (dotted line) and along a 45 ° -inclined nanotube (solid curve).

Fig. 7
Fig. 7

Evolution in time of the y component of the electric field around a 0.4 eccentricity ellipse at (a)  0 ° inclination (Media 1) and (b)  45 ° inclination (Media 2). Time progresses from left to right and from top down. Each frame represents a square area of 200 nm × 200 nm . Asterisks indicate the angular positions of one lobe of the rotating field.

Fig. 8
Fig. 8

Comparison between NRR and hybridization models by calculation of normalized resonant frequency ω / ω p as a function of radii ratio R 1 / R 2 of nanotubes with different thicknesses. Dashed curve represents the PH model [12]. Marked solid curves represent the NRR model at different thicknesses of the nanotube: 4 (triangles), 6 (rhombuses), 8 (circles), 10  (crosses), and 12 nm (squares).

Fig. 9
Fig. 9

Determination of resonant FSWs (vertical dashed lines) for three different nanotubes having the same radii ratios: (a) NRR model; (b) FDTD. Solid curves, labeled by numbers, correspond to 1, R 1 = 6 nm , R 2 = 10 nm , t = 4 nm ; 2, R 1 = 9 nm , R 2 = 15 nm , t = 6 nm ; 3, R 1 = 12 nm , R 2 = 20 nm , t = 8 nm . Horizontal dashed lines indicate the effective circumferences (perimeters) of the nanotubes.

Fig. 10
Fig. 10

Comparison between resonant FSWs calculated with the dispersion relation of the NRR model (solid curve) and those extracted from the scattering spectra of [10] (filled squares).

Tables (1)

Tables Icon

Table 1 Resonant Free-Space Wavelengths: the Hybridization and NRR Models versus FDTD Simulation Data

Equations (7)

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e 4 k 1 a = ( k 1 / ε 1 + k 2 / ε 2 k 1 / ε 1 k 2 / ε 2 ) ( k 1 / ε 1 + k 3 / ε 3 k 1 / ε 1 k 3 / ε 3 ) ,
λ s p = 2 π / Re β ,
d = 1 / Im β .
P eff ( λ ) = 2 π R 1 Re [ n 3 ( λ ) ] + R 2 Re [ n 2 ( λ ) ] Re [ n 3 ( λ ) ] + Re [ n 2 ( λ ) ] ,
λ s p ( λ res ) = P eff ( λ res ) / m = 2 π m R 1 Re [ n 3 ( λ res ) ] + R 2 Re [ n 2 ( λ res ) ] Re [ n 3 ( λ res ) ] + Re [ n 2 ( λ res ) ] ,
ε 1 ( ω ) = 1 ω p 2 ω 2 ,
λ res = 8 π c ω p [ 1 ± ( R 1 R 2 ) m ] 1 2 ,

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