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  • Editor: Anthony J. Campillo
  • Vol. 32, Iss. 12 — Jun. 15, 2007
  • pp: 1623–1625
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Design of plasmonic nanoantennae for enhancing spontaneous emission

Lavinia Rogobete, Franziska Kaminski, Mario Agio, and Vahid Sandoghdar  »View Author Affiliations


Optics Letters, Vol. 32, Issue 12, pp. 1623-1625 (2007)
http://dx.doi.org/10.1364/OL.32.001623


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Abstract

We apply two- and three-dimensional numerical calculations to study optical nanoantennae made of two coupled gold nanostructures, enclosing a single emitter in their gap. We show that, using structures manufacturable with today’s nanotechnology, it is possible to increase the radiative decay rate by three orders of magnitude while keeping a quantum efficiency larger than 80% in the near-infrared regime. We examine the competition between the radiative and nonradiative processes in the presence of the antennae as a function of wavelength and antenna geometry. Our results hold great promise for improving the quantum efficiency of poor emitters such as silicon nanocrystals or carbon nanotubes.

© 2007 Optical Society of America

The past two decades have seen tremendous progress in the control of the radiative decay rate (γr) of emitters embedded in solid microresonators [1

1. J.-M. Gérard and B. Gayral, in Confined Photon Systems: Fundamentals and Applications, H. Benisty, J.-M. Gérard, R. Houdré, J. Rarity, and C. Weisbuch, eds. (Springer, 1999), pp. 331–351. [CrossRef]

]. These systems occupy spaces larger than the emission wavelength and are therefore not suitable for applications requiring high densities. Advances in nano-optics have shown, however, that nanoparticles can also modify the spontaneous emission rate [2

2. H. Schniepp and V. Sandoghdar, Phys. Rev. Lett. 89, 257403 (2002). [CrossRef] [PubMed]

, 3

3. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006). [CrossRef] [PubMed]

, 4

4. R. Ruppin, J. Chem. Phys. 76, 1681 (1982). [CrossRef]

, 5

5. L. A. Blanco and F. J. Carcía de Abajo, Phys. Rev. B 69, 205414 (2004). [CrossRef]

, 6

6. J. R. Lakowicz, Anal. Biochem. 337, 171 (2005). [CrossRef] [PubMed]

]. In particular, metallic nanostructures are desirable because they can also enhance the excitation rate of an emitter close to them via their strong near fields [7

7. M. Moskovits, Rev. Mod. Phys. 57, 783 (1985). [CrossRef]

]. We have recently shown that a small metallic nanoparticle can be considered as a subwavelength dipolar antenna that modifies both the excitation and fluorescence of the emitter [3

3. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006). [CrossRef] [PubMed]

]. Other groups have worked to extend the conventional antenna designs to the nanometer scale [8

8. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, Phys. Rev. Lett. 94, 017402 (2005). [CrossRef] [PubMed]

, 9

9. P. Mühlschlegel, H.-J. Eisler, O. J. M. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005). [CrossRef] [PubMed]

]. Nevertheless, since metals are lossy in the visible range, at very short distances the nonradiative decay rate γnr dominates and the quantum efficiency η=γr(γr+γnr) drops [3

3. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006). [CrossRef] [PubMed]

, 4

4. R. Ruppin, J. Chem. Phys. 76, 1681 (1982). [CrossRef]

, 10

10. M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, Appl. Phys. Lett. 85, 3863 (2004). [CrossRef]

]. Thus, very strong enhancements of the excitation field might seem to be contradictory to a large enhancement of the emission rate.

In this Letter we numerically study the modifications of γr and η for an emitter very close to gold nanoantennae where it also experiences a substantial near-field enhancement of the excitation [7

7. M. Moskovits, Rev. Mod. Phys. 57, 783 (1985). [CrossRef]

]. We start by using a two-dimensional (2D) model based on the boundary-integral equations method to identify key issues that govern the influence of a nanoantenna on γr and γnr [11

11. L. Rogobete and C. Henkel, Phys. Rev. A 70, 063815 (2004). [CrossRef]

]. We then apply three-dimensional (3D) finite-difference time-domain (FDTD) calculations to investigate the system of choice quantitatively. In all following calculations, we consider the emitter and the antenna to be embedded in a homogeneous medium of refractive index equal to 1.7.

Let us begin by examining the elementary system of an emitter coupled to a gold nanosphere [3

3. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006). [CrossRef] [PubMed]

, 4

4. R. Ruppin, J. Chem. Phys. 76, 1681 (1982). [CrossRef]

]. The dotted curve in the inset of Fig. 1a shows the sphere scattering cross section (usually referred to as its plasmon spectrum) calculated for a plane wave illumination. The solid and dashed curves display the wavelength dependence of γr and γnr normalized to the unperturbed decay rate γr0 (i.e., in the absence of nanostructures) for an emitter–sphere separation of 3nm. Both γr and γnr are enhanced around the plasmon resonance, but γnr clearly dominates. The main part of Fig. 1a displays γr and γnr at the spectral maximum of γr as a function of the emitter–sphere separation, displaying the distance-dependent competition between them. This is indeed what has been pointed out in the literature for truncated tips [10

10. M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, Appl. Phys. Lett. 85, 3863 (2004). [CrossRef]

] and spherical particles [12

12. R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006). [CrossRef]

]. In what follows, we show that this is not a general rule and that it is possible to design antenna architectures that strongly enhance γr and at the same time maintain a large η.

A central idea in our work is to minimize the dissipation of energy into the antenna. Here we exploit the property that the plasmon resonance of a gold nanorod can be sharper than that of a gold nanosphere of the same volume [13

13. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]

] because the resonance corresponding to the longer axis of the particle shifts to the near-infrared region where the imaginary part of the dielectric function is smaller [14

14. CRC Handbookof Chemistry and Physics, 87th ed., http://www.hbcpnetbase.com (2006).

]. Inspired by this phenomenon and by the fact that the enhancement of the excitation field is expected to be strong at the apex of an elongated particle [7

7. M. Moskovits, Rev. Mod. Phys. 57, 783 (1985). [CrossRef]

, 15

15. J. Gersten and A. Nitzan, J. Chem. Phys. 75, 1139 (1981). [CrossRef]

], we have considered the emission of a dipole close to an elliptical particle. As shown in the inset of Fig. 1b, we see that, although both γr and γnr experience a substantial enhancement, γr overshadows the nonradiative losses at the plasmon resonance of the long axis. The main part of Fig. 1b plots the distance dependence of γr and γnr at the long-axis plasmon resonance, illustrating that in this case γr dominates for all separations larger than 3nm [16

16. Purely electromagnetic calculations are known to become inaccurate at very small distances of the order of 1nm due to local field effects so that we do not consider closer emitter–metal separations.

]. We note that in this work we do not discuss the strong quenching observed at shorter wavelengths, which we attribute to the excitation of higher multipoles [17

17. L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.

].

Having demonstrated the importance of the choice of the spectral domain and thus the design of the particle geometry, we now explore the possibility of even further enhancing γr. Here we are guided by the fact that under identical illumination conditions, the electric field between two nanostructures can be much stronger than the near field of a single structure [18

18. P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci. 110, 189 (1981). [CrossRef]

]. Using the reciprocity theorem [19

19. R. Carminati, M. Nieto-Versperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998). [CrossRef]

], we argue that also the emission of a dipole could undergo a larger enhancement between two metallic nanoparticles than in front of one. This has indeed been verified for two nanospheres [5

5. L. A. Blanco and F. J. Carcía de Abajo, Phys. Rev. B 69, 205414 (2004). [CrossRef]

]. In Fig. 2 , we extend this concept to other geometries. The curves with the diamonds replot the data of a single ellipse from Fig. 1b for comparison. The solid and dashed curves with the upward-pointing triangles depict the wavelength dependence of γr and η for an emitter placed in the middle of two identical ellipses with a gap of 6nm. We find that γr has been enhanced by about 1000 times. Furthermore, although more metal surrounds the emitter in the double-ellipse antenna, η reaches a higher value of 0.9. Note that the plasmon resonance of the new structure has shifted to longer wavelengths and has become broader because of the near-field coupling of the two ellipses [18

18. P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci. 110, 189 (1981). [CrossRef]

, 20

20. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, Opt. Commun. 220, 137 (2003). [CrossRef]

]. We emphasize that this broadening stems from a larger polarizability and therefore a stronger scattering cross section; it should not be confused with a dissipative broadening.

Next, we compare the performance of the double-ellipse antenna with the bow-tie antenna, which has been studied recently [8

8. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, Phys. Rev. Lett. 94, 017402 (2005). [CrossRef] [PubMed]

]. To isolate the role of the antenna shape in its performance, we keep the overall structure area the same and maintain a constant separation between the emitter and its nearest metallic surface. As seen from the curves with squares in Fig. 2, γr and η are both less favorable than for the double-ellipse geometry at the respective plasmon resonances. However, the situation changes when we elongate the triangles to have a tip radius of curvature of the order of 1nm. The curves with the downward-pointing triangles reveal an enhancement of γr in excess of 1700 while keeping η0.9. This substantial improvement is unfortunately not within reach of the current technology because of the difficulty in fabricating sharp corners and in avoiding problems such as tip snipping [21

21. E. Hao, R. C. Bailey, G. C. Schatz, J. T. Hupp, and S. Li, Nano Lett. 4, 327 (2004). [CrossRef]

]. Moreover, very sharp corners also pose a serious challenge in theoretical studies [16

16. Purely electromagnetic calculations are known to become inaccurate at very small distances of the order of 1nm due to local field effects so that we do not consider closer emitter–metal separations.

]. As a result, in what follows we identify the double-ellipse structure as the ideal compromise for obtaining very large emission enhancement.

To evaluate the antenna performance more quantitatively, we have computed γr, γnr, and η using the 3D FDTD method. Details on these calculations and numerical tests are given in [17

17. L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.

, 22

22. F. Kaminski, V. Sandoghdar, and M. Agio, J. Comput. Theor. Nanosci. 4, 635 (2007).

]. We choose prolate spheroids with long and short axes of 120 and 38nm, respectively, with a gap of width d (see the inset in Fig. 3 ). This places the plasmon resonance peak wavelength of a single spheroid around 1μm. We point out that the choice of the particle size and therefore the spectral optimum of the antenna allows a certain degree of tunability [17

17. L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.

]. However, one has to bear in mind that particles that are too small lead to strong absorption [15

15. J. Gersten and A. Nitzan, J. Chem. Phys. 75, 1139 (1981). [CrossRef]

], whereas particles that are too large result in higher multipole moments [23

23. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

] and thus less efficient polarization and mode overlap with the emitter. Figure 3 shows the normalized radiative decay rate as a function of d for an emitter at the center of the gap and oriented along the antenna axis. The enhancement is of the order of 5000 for d=10nm and over 800 for d=20nm [24

24. Although our FDTD mesh pitch was as small as 1nm, the results for d=10nm might be inaccurate due to the slow convergence of the calculation [17].

]. In fact, even for a large antenna gap of 50nm, there remains a considerable enhancement of 80 times. We obtain η0.8 and have verified that it depends weakly on d and the wavelength [17

17. L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.

]. As shown in the inset, we have also confirmed that the emission pattern remains dipolar.

In summary, we have achieved a very strong enhancement of spontaneous emission with minimal suffering from nonradiative losses. The key design principles have been, first, to tailor the geometry such that the plasmon resonance of the antenna lies in a favorable spectral region for minimizing dissipation in the metal. Second, we have chosen elongated objects to benefit from strong near fields at sharp corners. Third, we have selected antennas with dipolar resonances and have chosen the emitter orientation such that its electric dipole moment is aligned with that of the antenna. Fourth, we have taken into account realistic fabrication constraints concerning structure sharpness and antenna gap in the existing nanofabrication techniques such as electron beam lithography or focused ion beam milling. The details of these studies will be published separately [17

17. L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.

]. An exhaustive exploration of parameters such as the material, size, and shape of the antenna as well as the position, orientation, and transition wavelength of the emitter could yield even better results and remains the subject of future studies.

From the application point of view, a stronger enhancement of γr compared with γnr allows improvement of the quantum efficiency of poor emitters [6

6. J. R. Lakowicz, Anal. Biochem. 337, 171 (2005). [CrossRef] [PubMed]

] such as silicon nanocrystals [25

25. J. S. Biteen, D. Pacifici, N. S. Lewis, and H. A. Atwater, Nano Lett. 5, 1768 (2005). [CrossRef] [PubMed]

] or nanotubes [26

26. M. J. O’Connell, S. M. Bachilo, C. B. Huffman, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, J. Ma, R. H. Hauge, R. B. Weisman, and R. E. Smalley, Science 297, 593 (2002). [CrossRef] [PubMed]

] and could provide a handle on the photophysics of emitters in general. Furthermore, even for emitters with high quantum efficiency a larger emission rate permits a higher degree of light extraction by postponing saturation to higher intensities. These phenomena could be exploited for a range of light-emitting devices such as displays or pigments based on composite material made of a carefully fabricated combination of nanoantennae and emitters.

M. Agio thanks A. Mohammadi for help with FDTD calculations and C. Henkel for discussions. This work was financed by the ETH Zurich initiative on Composite Doped Metamaterials.

Fig. 1 Normalized 2D radiative (solid curves) and non-radiative (dashed curves) decay rates for an emitter coupled to a gold sphere (diameter=24.5nm) (a) and to an ellipse (long axis=60nm, short axis=10nm) (b). The emission wavelength is tuned to λ=535nm for the sphere and λ=770nm for the ellipse. The particles have the same area. Insets: normalized decay rates as a function of wavelength for a particle–emitter distance of 3nm. The scattering cross sections of the systems are also plotted (dotted curves) to show their plasmon resonances for comparison. The emitter is oriented as shown in the graph.
Fig. 2 2D-calculated normalized radiative decay rates (solid curves) and quantum efficiencies (dashed curves) for different gold nanoantennae. All structures have the same area (A491nm2 per particle) and a gap width of 6nm. The emitter is placed at the center of the gap and oriented as shown in the graph.
Fig. 3 3D-calculated normalized radiative decay rates for nanoantennae made of two prolate gold spheroids (long axis 120nm, short axes 38nm). The legend gives the gap width d in nm. The emitter is placed at the center of the gap and oriented along the nanoantenna. The inset shows that the emission pattern remains dipolar.
1.

J.-M. Gérard and B. Gayral, in Confined Photon Systems: Fundamentals and Applications, H. Benisty, J.-M. Gérard, R. Houdré, J. Rarity, and C. Weisbuch, eds. (Springer, 1999), pp. 331–351. [CrossRef]

2.

H. Schniepp and V. Sandoghdar, Phys. Rev. Lett. 89, 257403 (2002). [CrossRef] [PubMed]

3.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006). [CrossRef] [PubMed]

4.

R. Ruppin, J. Chem. Phys. 76, 1681 (1982). [CrossRef]

5.

L. A. Blanco and F. J. Carcía de Abajo, Phys. Rev. B 69, 205414 (2004). [CrossRef]

6.

J. R. Lakowicz, Anal. Biochem. 337, 171 (2005). [CrossRef] [PubMed]

7.

M. Moskovits, Rev. Mod. Phys. 57, 783 (1985). [CrossRef]

8.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, Phys. Rev. Lett. 94, 017402 (2005). [CrossRef] [PubMed]

9.

P. Mühlschlegel, H.-J. Eisler, O. J. M. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005). [CrossRef] [PubMed]

10.

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, Appl. Phys. Lett. 85, 3863 (2004). [CrossRef]

11.

L. Rogobete and C. Henkel, Phys. Rev. A 70, 063815 (2004). [CrossRef]

12.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006). [CrossRef]

13.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]

14.

CRC Handbookof Chemistry and Physics, 87th ed., http://www.hbcpnetbase.com (2006).

15.

J. Gersten and A. Nitzan, J. Chem. Phys. 75, 1139 (1981). [CrossRef]

16.

Purely electromagnetic calculations are known to become inaccurate at very small distances of the order of 1nm due to local field effects so that we do not consider closer emitter–metal separations.

17.

L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.

18.

P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci. 110, 189 (1981). [CrossRef]

19.

R. Carminati, M. Nieto-Versperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998). [CrossRef]

20.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, Opt. Commun. 220, 137 (2003). [CrossRef]

21.

E. Hao, R. C. Bailey, G. C. Schatz, J. T. Hupp, and S. Li, Nano Lett. 4, 327 (2004). [CrossRef]

22.

F. Kaminski, V. Sandoghdar, and M. Agio, J. Comput. Theor. Nanosci. 4, 635 (2007).

23.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

24.

Although our FDTD mesh pitch was as small as 1nm, the results for d=10nm might be inaccurate due to the slow convergence of the calculation [17].

25.

J. S. Biteen, D. Pacifici, N. S. Lewis, and H. A. Atwater, Nano Lett. 5, 1768 (2005). [CrossRef] [PubMed]

26.

M. J. O’Connell, S. M. Bachilo, C. B. Huffman, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, J. Ma, R. H. Hauge, R. B. Weisman, and R. E. Smalley, Science 297, 593 (2002). [CrossRef] [PubMed]

OCIS Codes
(020.0020) Atomic and molecular physics : Atomic and molecular physics
(270.0270) Quantum optics : Quantum optics
(290.5850) Scattering : Scattering, particles

ToC Category:
Quantum Optics

History
Original Manuscript: February 15, 2007
Manuscript Accepted: March 12, 2007
Published: June 5, 2007

Citation
Lavinia Rogobete, Franziska Kaminski, Mario Agio, and Vahid Sandoghdar, "Design of plasmonic nanoantennae for enhancing spontaneous emission," Opt. Lett. 32, 1623-1625 (2007)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-32-12-1623


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References

  1. J.-M. Gérard and B. Gayral, in Confined Photon Systems: Fundamentals and Applications, H.Benisty, J.-M.Gérard, R.Houdré, J.Rarity, and C.Weisbuch, eds. (Springer, 1999), pp. 331-351. [CrossRef]
  2. H. Schniepp and V. Sandoghdar, Phys. Rev. Lett. 89, 257403 (2002). [CrossRef] [PubMed]
  3. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006). [CrossRef] [PubMed]
  4. R. Ruppin, J. Chem. Phys. 76, 1681 (1982). [CrossRef]
  5. L. A. Blanco and F. J. Carcía de Abajo, Phys. Rev. B 69, 205414 (2004). [CrossRef]
  6. J. R. Lakowicz, Anal. Biochem. 337, 171 (2005). [CrossRef] [PubMed]
  7. M. Moskovits, Rev. Mod. Phys. 57, 783 (1985). [CrossRef]
  8. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, Phys. Rev. Lett. 94, 017402 (2005). [CrossRef] [PubMed]
  9. P. Mühlschlegel, H.-J. Eisler, O. J. M. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005). [CrossRef] [PubMed]
  10. M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, Appl. Phys. Lett. 85, 3863 (2004). [CrossRef]
  11. L. Rogobete and C. Henkel, Phys. Rev. A 70, 063815 (2004). [CrossRef]
  12. R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006). [CrossRef]
  13. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]
  14. CRC Handbook of Chemistry and Physics, 87th ed., http://www.hbcpnetbase.com (2006).
  15. J. Gersten and A. Nitzan, J. Chem. Phys. 75, 1139 (1981). [CrossRef]
  16. Purely electromagnetic calculations are known to become inaccurate at very small distances of the order of 1 nm due to local field effects so that we do not consider closer emitter-metal separations.
  17. L. Rogobete, F. Kaminski, A. Mohammadi, M. Agio, and V. Sandoghdar, manuscript in preparation.
  18. P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci. 110, 189 (1981). [CrossRef]
  19. R. Carminati, M. Nieto-Versperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998). [CrossRef]
  20. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, Opt. Commun. 220, 137 (2003). [CrossRef]
  21. E. Hao, R. C. Bailey, G. C. Schatz, J. T. Hupp, and S. Li, Nano Lett. 4, 327 (2004). [CrossRef]
  22. F. Kaminski, V. Sandoghdar, and M. Agio, J. Comput. Theor. Nanosci. 4, 635 (2007).
  23. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).
  24. Although our FDTD mesh pitch was as small as 1 nm, the results for d=10 nm might be inaccurate due to the slow convergence of the calculation .
  25. J. S. Biteen, D. Pacifici, N. S. Lewis, and H. A. Atwater, Nano Lett. 5, 1768 (2005). [CrossRef] [PubMed]
  26. M. J. O'Connell, S. M. Bachilo, C. B. Huffman, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, J. Ma, R. H. Hauge, R. B. Weisman, and R. E. Smalley, Science 297, 593 (2002). [CrossRef] [PubMed]

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