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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 17 — Aug. 17, 2009
  • pp: 14967–14976
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Effects of vertex truncation of polyhedral nanostructures on localized surface plasmon resonance

W. Y. Ma, J. Yao, H. Yang, J. Y. Liu, F. Li, J. P. Hilton, and Q. Lin  »View Author Affiliations


Optics Express, Vol. 17, Issue 17, pp. 14967-14976 (2009)
http://dx.doi.org/10.1364/OE.17.014967


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Abstract

Polyhedral nanostructures are widely used to enable localized surface plasmon resonance (LSPR). In practice, vertices of such structures are almost always truncated due to limitations of nanofabrication processes. This paper studies the effects of vertex truncation of polyhedral nanostructures on the characteristics of LSPR sensing. The optical properties and sensing performance of triangular nanoplates with truncated vertices are investigated using electrodynamics analysis and verified by experiment. The experimental results correlated with simulation analysis demonstrate that the fabricated triangular nanoplate array has a truncation ratio, defined as the length of truncation along an edge of the triangle over the edge length, of approximately 12.8%. This significantly influences optical properties of the nanostructures, resulting in poorer sensing performance. These insights can be used to guide the design and fabrication of nanostructures for high performance LSPR sensors.

© 2009 OSA

1. Introduction

Noble metal nanostructures are increasingly enabling practically important applications such as localized surface plasmon resonance (LSPR) [1

1. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [PubMed]

3

3. F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007). [PubMed]

]. In LSPR, a nanostructure absorbs and scatters light, resulting in an extinction spectrum (also called LSPR spectrum) that can be measured by a spectrometer. The LSPR spectrum of a nanostructure depends strongly on its size, shape and composition [4

4. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).

,5

5. M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” J. Phys. Chem. B 109(46), 21556–21565 (2005).

], and exhibits high sensitivity to environmental changes. Due to these characteristics, nanostructures have potential utility in immunoassays and other biochemical sensing applications [6

6. P. K. Jain, I. H. El-sayed, and M. A. El-sayed, “Au nanoparticles target cancer,” Nano Today 2(1), 18–29 (2007).

8

8. K. A. Willets and R. P. V. Duyne, “Localized surface plasmon spectroscopy and sensing,” Annu. Rev. Chem. 58(1), 267–297 (2007).

]. Current research in this area is focusing on the use of various non-spherical nanostructures, such as nanorods, nanocubes, rhombic and triangular nanostructures [9

9. C. D. Chen, S. F. Cheng, L. K. Chau, and C. R. C. Wang, “Sensing capability of the localized surface plasmon resonance of gold nanorods,” Biosens. Bioelectron. 22(6), 926–932 (2007).

13

13. G. H. Chan, J. Zhao, G. C. Schatz, and R. P. V. Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Triangular Aluminum Nanoparticles,” J. Phys. Chem. C 112(36), 13958–13963 (2008).

], all of which are shapes with sharp vertices and edges.

The work presented herein utilizes a typical two-dimensional array of silver triangular nanoplates to study the impact of truncated vertices on LSPR sensing with polyhedral nanostructures. Numerical models of nanostructures with different truncation ratios are created and the finite difference time domain (FDTD) method is employed to explore the effect of truncated vertices on the electromagnetic field of the nanostructures. Following this analysis, experimental results are generated using an array of triangular nanoplates fabricated with NSL. These results are correlated to the previously generated model to show how truncated vertices of nanostructures predictably degrade their optical properties.

2. Numerical models and electrodynamics results

In this work, the idealized structure is represented as a triangular plate with an equilateral base (consisting of three straight edges and three vertices), which has a height h = 50 nm and an edge length of a = 156 nm, as shown in Fig. 1(a)
Fig. 1 Triangular nanoplate structures. (a) An idealized triangular nanoplate. (b) A triangular nanoplate with truncated vertices. (c) The vertex truncation is assumed to be formed by a symmetric circular arc tangent to each of the two edges. The truncation length, t, is defined as the distance between the vertex of the triangle and the point of tangency.
. Following vertex truncation, the nanostructure exhibits truncated vertices [16

16. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [PubMed]

] as shown in Fig. 1 (b). FDTD calculations have been performed for arrays of triangular plates with truncated vertices arranged in a hexagonal form with a period of 520 nm, the vertex truncation is assumed to be formed by a symmetric circular arc tangent to each of two edges of the triangle. The truncation length, t, defined as the distance between a vertex of the triangle and the closest point of tangency, has a value of 0, 5, 10, 12, 15, 20, or 25 nm. The extent of vertex truncation is characterized by the truncation ratio, which is defined as σ = t/a.

The transverse magnetic (TM) Maxwell curl equations, shown below in Eqs. (1) – (3), are solved using the FDTD numerical method to generate results [17

17. A. Taflove, and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, (Arthech House, London, 2000).

]:
Hyt=1μ0(ExzEzx),
(1)
Ext=1εHyz,
(2)
Ezt=1εHyx,
(3)
where H and E are the magnetic and the electric field, respectively, and their subscript x, y and z denote the three electromagnetic field components. Here, μ0is the magnetic permeability in vacuum, ε=ε0εr is the permittivity of the nanostructure material, with ε0 the permittivity of the vacuum, and εr the materials’ relative permittivity. For silver in this work, the values of εr as a function of wavelength are obtained from Palik [18

18. E. D. Palik, Handbook of optical constants of solids III, (Academic Press, 1998).

]. The extinction cross section of the array of nanoparticles is defined as:
Cext(ω)=Cscat(ω)+Cabs(ω)=Pscat+PabsIinc(ω),
(4)
The power transmission parameter P is obtained in terms of the Poynting vector:P=E×H, and the subscripts ext, scat, abs and inc represent extinction, scatter, absorption and incidence, respectively. In our calculations, the substrate is assumed to be glass with an area slightly larger than that of the target nanoplates array and a refractive index of 1.51. The surrounding medium around the nanostructures is assumed to be air with a refractive index of 1.0, and the light source is modeled as non-polarized in order to imitate natural light.

According to the calculated extinction spectra presented in Fig. 2 (a)
Fig. 2 Calculation results of a triangular nanoplates array with a truncated length t = 0, 5, 10, 12, 15, 20 and 25 nm, with (a) Extinction spectra distributions and (b) peak wavelength distribution extracted from (a)
, the localized surface plasmon resonance extinction spectrum exhibits a peak at about 741 nm for the idealized structure (σ = 0). As indicated in Fig. 2 (b), the peak wavelength of the LSPR spectrum shifts from 741 nm to 650 nm gradually when the truncation ratio σ varies from 0 to 16%. This phenomenon is caused by the retardation effects of the electromagnetic field across the particles as the characteristic size of the nanoparticles decreases [19

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

].

The extinction efficiency in Fig. 2(a) is based on the surface enhanced mechanism [19

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

,20

20. C. Bohren, and D. Huffman, Absorption and Scattering of Light by Small Particles, (J. Wiley, New York, 1983).

], which intensifies the local electric field (E-field) at the vertices of the nanostructures. Comparing Fig. 3(a)
Fig. 3 Field enhancement of local E-fields of nanostructures with a truncation ratioσof 12.8% (a) and the idealized structure (b). The E-field enhancement profile is presented along the hemline for a truncated structure (c) and the idealized structure (d). The E-fields are excited at their LSPR peak wavelength of 669 nm and 741 nm, respectively.
with Fig. 3(b), it is apparent that the E-field distribution of structures with sharp vertices is more localized than that of the truncated structure with a truncation ratio σ of 12.8%, and the maximal field enhancement of the idealized structure is about 10 times that of the truncated one, as shown in Figs. 3(c) and 3(d). Apparently, it is the less effective concentration of the E-field near the vertices that contributes to the decrease in extinction efficiency of truncated structures (Fig. 2(a)).

To further investigate the influence of truncated vertices on the sensitivity of the optical nanostructures, we have also performed the FDTD calculations to explore the refractive index sensitivity of the nanostructures with different truncation ratios. The refractive index sensitivity for a LSPR sensor is defined as:m=Δλp/Δn, whereΔλpandΔndenotes peak wavelength shift in a different medium and the corresponding refractive index change [21

21. M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. V. Duyne, “Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers,” J. Am. Chem. Soc. 123(7), 1471–1482 (2001).

]. For an idealized structure, the peak wavelength shifts higher when the refractive index increases, and the refractive index sensitivity m is as large as 434 nm/RIU (refractive index unit) (Fig. 4(a)
Fig. 4 (a) Extinction spectra of an idealized structure in different media and the obtained refractive index curve and (b) Refractive index sensitivity distribution of truncated structures with truncation ratio σ = 0, 3.2%, 6.4%, 7.7%, 9.6%, 12.8%, or 16%.
). The refractive index sensitivity decreases strongly when σ increases from 0 to 3.2%, as shown in Fig. 4 (b). This indicates that nanoparticles with sharp vertices exhibit high sensitivity while the performance of nanoparticles with slightly rounded vertices is much poorer. When σ varies from 6.4% to 16%, the refractive indices fall from 336 nm/RIU to 250 nm/RIU. It should be noted that the refractive index sensitivity shown in Fig. 4(b) has an almost identical relationship to truncation ratio as the extinction spectra peak wavelength as shown in Fig. 2(b). This can be explained using quasistatic theory: if the real part of the dielectric functionε1is approximated as varying linearly with wavelength, the refractive index sensitivity m is approximately linear with the extinction spectra peak wavelength [5

5. M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” J. Phys. Chem. B 109(46), 21556–21565 (2005).

].

The overall LSPR sensing performance of a nanostructure is evaluated by the figure of merit (FOM), which is defined as [22

22. L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5(10), 2034–2038 (2005). [PubMed]

]:
FOM=m/FWHM,
(5)
where FWHM is the full-width at half-maximum of the LSPR spectrum in air (n = 0). Unlike the refractive index sensitivity, which is an extrinsic parameter, FWHM is intrinsically determined by the shape of a nanostructure and is used to evaluate the resolution factor. The calculated FOM values decline as the truncation ratio increases, as shown by the fitted curve in Fig. 5
Fig. 5 Figure of merit distributions for different truncation ratios
. According to this relationship, whenσ10%, the FOM is lower than 2.0%, and this results in much poorer sensing performance by the nanostructures array. The small variations of FOM values (hollow circles in Fig. 5) are probably caused by the random energy level splitting of free electrons in metals [23

23. V. F. Weisskopf, “Recent development in the theory of the electron,” Rev. Mod. Phys. 21(2), 305–315 (1949).

].

3. Experiment results and analysis

The LSPR-based extinction spectra of the fabricated Ag nanostructures were measured both in clean air (n≈1.0) and absolute alcohol (n≈1.37) at room temperature (20°C) by an UV-vis-NIR spectrometer (QE65000, Ocean Optics inc. America), with associated data processing software. For n = 1.0, the peak wavelength was measured to be approximately 670 nm (Fig. 7(a)
Fig. 7 Extinction spectra obtained from simulations and experiments: (a) in air, and (b) in ethanol.
), which closely matches the calculated result for a truncated structure with a truncation ratio σ of 12.8%. This result differs by approximately 71 nm with that of the idealized structure as shown in Fig. 7(a). The measured extinction spectrum shows good agreement for n = 1.37, with results similar to those calculated forσ=12.8%. Both the SEM image and the measured extinction spectra confirm that the fabricated nanostructure has an average truncation ratio approximately of 12.8%, and that truncated vertices strongly influence the optical properties of nanostructures for a LSPR sensor.

The refractive index sensitivity for the fabricated structure can be calculated as m=(756 nm670 nm)/(0.37 RIU)232.4 nm/RIU, and the FWHM of the LSPR spectrum in air is about 160 nm (see Fig. 7(a)). Therefore, FOM ≈1.45 RIU−1as calculated. This is much smaller than that of the idealized structure, 2.66RIU−1, indicating that the performance of the fabricated LSPR sensor with truncated corners is greatly reduced. It should be noted that the test FOM of 1.45 RIU−1has a small difference with that of the simulation result for σ=12.8%of 1.73 RIU−1. This can be attributed to two factors. First, the characteristic sizes and the shapes of fabricated nanostructures array are not as homogeneous as considered in the simulation. As a result, the FWHM of the extinction spectrum in air for the fabricated nanostructures is a little wider than the simulation results. Second, the impurities in the medium and manipulation errors significantly influence the test extinction spectra, especially the spectra in ethanol medium, which further influence the test refractive index sensitivity. The sensing performance of the fabricated nanostructures will be further improved when the sample is measured in N2 medium without dust and vapor.

4. Conclusion

This work presents a study on the effects of truncated vertices on the optical properties of polyhedral nanostructures by combined simulations and experiments. Both the electrodynamics analysis and the experimental results demonstrate the great influence of truncated vertices on the performance of a LSPR sensor. Simulation results indicate that the peak wavelength of a LSPR spectrum shifts blue and the FOM of sensing performance becomes poorer when the truncation ratio increases, as a result of the depressed E-field at the truncated vertices. The corresponding fabricated triangular nanoplate array was shown to have a truncation ratio of 12.8%, which is consistent with the simulation results. This study is expected to provide guidlines for both designing and fabricating a nanostructure for LSPR sensing.

Acknowledgements

This work was supported by the Natural Science Foundation of China (60736037) and the Opening Project of Key Laboratory of Low Dimensional Quantum Structures and Quantum Control (Hunan Normal University), Ministry of Education, (QSQC0909).

References and Links

1.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [PubMed]

2.

M. Li, Z. S. Zhang, X. Zhang, K. Y. Li, and X. F. Yu, “Optical properties of Au/Ag core/shell nanoshuttles,” Opt. Express 16(18), 14288–14293 (2008). [PubMed]

3.

F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007). [PubMed]

4.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).

5.

M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” J. Phys. Chem. B 109(46), 21556–21565 (2005).

6.

P. K. Jain, I. H. El-sayed, and M. A. El-sayed, “Au nanoparticles target cancer,” Nano Today 2(1), 18–29 (2007).

7.

S. L. Zhu, F. Li, C. L. Du, and Y. Q. Fu, “A localized surface plasmon resonance nanosensor based on rhombic Ag nanoparticle array,” Sens. Actuators B Chem. 134(1), 193–198 (2008).

8.

K. A. Willets and R. P. V. Duyne, “Localized surface plasmon spectroscopy and sensing,” Annu. Rev. Chem. 58(1), 267–297 (2007).

9.

C. D. Chen, S. F. Cheng, L. K. Chau, and C. R. C. Wang, “Sensing capability of the localized surface plasmon resonance of gold nanorods,” Biosens. Bioelectron. 22(6), 926–932 (2007).

10.

J. Becker, I. Zins, A. Jakab, Y. Khalavka, O. Schubert, and C. Sönnichsen, “Plasmonic focusing reduces ensemble linewidth of silver-coated gold nanorods,” Nano Lett. 8(6), 1719–1723 (2008). [PubMed]

11.

J. M. McLellan, A. Siekkinen, J. Chen, and Y. Xia, “Comparison of the surface-enhanced Raman scattering on sharp and truncated silver nanocubes,” Chem. Phys. Lett. 427(1-3), 122–126 (2006).

12.

S. L. Zhu, C. L. Du, and Y. Q. Fu, “Fabrication and characterization of rhombic silver nanoparticles for biosensing,” Opt. Mater. 31(6), 769–774 (2009).

13.

G. H. Chan, J. Zhao, G. C. Schatz, and R. P. V. Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Triangular Aluminum Nanoparticles,” J. Phys. Chem. C 112(36), 13958–13963 (2008).

14.

B. J. Wiley, Y. Chen, J. M. McLellan, Y. Xiong, Z. Y. Li, D. Ginger, and Y. Xia, “Synthesis and optical properties of silver nanobars and nanorice,” Nano Lett. 7(4), 1032–1036 (2007). [PubMed]

15.

Y. Xia, N. J. Halas, and G. Editors, “Shape-Controlled Synthesis and Surface Plasmonic Properties of Metallic Nanostructures,” MRS Bull. 30, 338–348 (2005).

16.

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [PubMed]

17.

A. Taflove, and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, (Arthech House, London, 2000).

18.

E. D. Palik, Handbook of optical constants of solids III, (Academic Press, 1998).

19.

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

20.

C. Bohren, and D. Huffman, Absorption and Scattering of Light by Small Particles, (J. Wiley, New York, 1983).

21.

M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. V. Duyne, “Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers,” J. Am. Chem. Soc. 123(7), 1471–1482 (2001).

22.

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5(10), 2034–2038 (2005). [PubMed]

23.

V. F. Weisskopf, “Recent development in the theory of the electron,” Rev. Mod. Phys. 21(2), 305–315 (1949).

24.

C. L. Haynes and R. P. V. Duyne, “Nanosphere lithography: a versatile nanofabrication tool for studies of size-dependent nanoparticle optics,” J. Phys. Chem. B 105(24), 5599–5611 (2001).

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 1, 2009
Revised Manuscript: July 24, 2009
Manuscript Accepted: July 25, 2009
Published: August 10, 2009

Citation
W. Y. Ma, J. Yao, H. Yang, J. Y. Liu, F. Li, J. P. Hilton, and Q. Lin, "Effects of vertex truncation of polyhedral nanostructures on localized surface plasmon resonance," Opt. Express 17, 14967-14976 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14967


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References

  1. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [PubMed]
  2. M. Li, Z. S. Zhang, X. Zhang, K. Y. Li, and X. F. Yu, “Optical properties of Au/Ag core/shell nanoshuttles,” Opt. Express 16(18), 14288–14293 (2008). [PubMed]
  3. F. Hao, C. L. Nehl, J. H. Hafner, and P. Nordlander, “Plasmon resonances of a gold nanostar,” Nano Lett. 7(3), 729–732 (2007). [PubMed]
  4. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).
  5. M. M. Miller and A. A. Lazarides, “Sensitivity of metal nanoparticle surface plasmon resonance to the dielectric environment,” J. Phys. Chem. B 109(46), 21556–21565 (2005).
  6. P. K. Jain, I. H. El-sayed, and M. A. El-sayed, “Au nanoparticles target cancer,” Nano Today 2(1), 18–29 (2007).
  7. S. L. Zhu, F. Li, C. L. Du, and Y. Q. Fu, “A localized surface plasmon resonance nanosensor based on rhombic Ag nanoparticle array,” Sens. Actuators B Chem. 134(1), 193–198 (2008).
  8. K. A. Willets and R. P. V. Duyne, “Localized surface plasmon spectroscopy and sensing,” Annu. Rev. Chem. 58(1), 267–297 (2007).
  9. C. D. Chen, S. F. Cheng, L. K. Chau, and C. R. C. Wang, “Sensing capability of the localized surface plasmon resonance of gold nanorods,” Biosens. Bioelectron. 22(6), 926–932 (2007).
  10. J. Becker, I. Zins, A. Jakab, Y. Khalavka, O. Schubert, and C. Sönnichsen, “Plasmonic focusing reduces ensemble linewidth of silver-coated gold nanorods,” Nano Lett. 8(6), 1719–1723 (2008). [PubMed]
  11. J. M. McLellan, A. Siekkinen, J. Chen, and Y. Xia, “Comparison of the surface-enhanced Raman scattering on sharp and truncated silver nanocubes,” Chem. Phys. Lett. 427(1-3), 122–126 (2006).
  12. S. L. Zhu, C. L. Du, and Y. Q. Fu, “Fabrication and characterization of rhombic silver nanoparticles for biosensing,” Opt. Mater. 31(6), 769–774 (2009).
  13. G. H. Chan, J. Zhao, G. C. Schatz, and R. P. V. Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Triangular Aluminum Nanoparticles,” J. Phys. Chem. C 112(36), 13958–13963 (2008).
  14. B. J. Wiley, Y. Chen, J. M. McLellan, Y. Xiong, Z. Y. Li, D. Ginger, and Y. Xia, “Synthesis and optical properties of silver nanobars and nanorice,” Nano Lett. 7(4), 1032–1036 (2007). [PubMed]
  15. Y. Xia, N. J. Halas, and G. Editors, “Shape-Controlled Synthesis and Surface Plasmonic Properties of Metallic Nanostructures,” MRS Bull. 30, 338–348 (2005).
  16. S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008). [PubMed]
  17. A. Taflove, and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, (Arthech House, London, 2000).
  18. E. D. Palik, Handbook of optical constants of solids III, (Academic Press, 1998).
  19. U. Kreibig, and M. Vollmer, Optical Properties of Metal Clusters (Springer, Berlin, 1995).
  20. C. Bohren, and D. Huffman, Absorption and Scattering of Light by Small Particles, (J. Wiley, New York, 1983).
  21. M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. V. Duyne, “Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers,” J. Am. Chem. Soc. 123(7), 1471–1482 (2001).
  22. L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5(10), 2034–2038 (2005). [PubMed]
  23. V. F. Weisskopf, “Recent development in the theory of the electron,” Rev. Mod. Phys. 21(2), 305–315 (1949).
  24. C. L. Haynes and R. P. V. Duyne, “Nanosphere lithography: a versatile nanofabrication tool for studies of size-dependent nanoparticle optics,” J. Phys. Chem. B 105(24), 5599–5611 (2001).

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