Abstract

We present a theoretical model for describing light scattering from randomly distributed Au nanoparticles on a substrate, including the clustering effect. By using the finite-element Green’s function method and spherical harmonic basis functions, we are able to calculate the polarization-dependent reflectivity spectra of the system (modeled by randomly distributed nanoparticles coupled with clusters) efficiently and accurately. The calculated ellipsometric spectra of the system with clusters can adequately describe the experimental data for the whole frequency range. We find that the clustering effect leads to some prominent features in the low frequency range of the ellipsometric spectra, which are attributed to plasmonic resonances associated with the coupling of Au nanoparticles and clusters.

©2013 Optical Society of America

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References

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    [Crossref] [PubMed]
  4. A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B 104(50), 11965–11971 (2000).
    [Crossref]
  5. A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. 48(14), 957–960 (1982).
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    [Crossref]
  8. B. Kaplan, T. Novikova, A. De Martino, and B. Drévillon, “Characterization of bidimensional gratings by spectroscopic ellipsometry and angle-resolved Mueller polarimetry,” Appl. Opt. 43(6), 1233–1240 (2004).
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  9. H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
    [Crossref]
  10. S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
    [Crossref]
  11. D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
    [Crossref]
  12. G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
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    [Crossref]
  20. Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5(5), 1194–1197 (2008).
    [Crossref]
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  22. S. H. Hsu, Y. C. Chang, Y. C. Chen, P. K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18(2), 1310–1315 (2010).
    [Crossref] [PubMed]
  23. R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films 419(1-2), 124–136 (2002).
    [Crossref]
  24. G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
    [Crossref]
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  28. See for example, Fayyazuddin and Riazuddin, Quantum Mechanics (World Scientific, 1990), p. 368.

2011 (2)

2010 (1)

2009 (1)

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

2008 (2)

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5(5), 1194–1197 (2008).
[Crossref]

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

2007 (1)

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
[Crossref]

2006 (1)

2004 (2)

B. Kaplan, T. Novikova, A. De Martino, and B. Drévillon, “Characterization of bidimensional gratings by spectroscopic ellipsometry and angle-resolved Mueller polarimetry,” Appl. Opt. 43(6), 1233–1240 (2004).
[Crossref] [PubMed]

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

2002 (1)

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films 419(1-2), 124–136 (2002).
[Crossref]

2001 (2)

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

2000 (3)

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(33 Pt B), 4318–4324 (2000).
[Crossref] [PubMed]

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B 104(50), 11965–11971 (2000).
[Crossref]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B 61(11), 7722–7733 (2000).
[Crossref]

1999 (1)

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc. 121(39), 9208–9214 (1999).
[Crossref]

1997 (1)

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

1995 (1)

1991 (1)

1982 (1)

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. 48(14), 957–960 (1982).
[Crossref]

1975 (1)

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Aizpurua, J.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(33 Pt B), 4318–4324 (2000).
[Crossref] [PubMed]

Apell, P.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(33 Pt B), 4318–4324 (2000).
[Crossref] [PubMed]

Asano, S.

Bedeaux, D.

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

Booso, B.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

Brus, L.

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B 104(50), 11965–11971 (2000).
[Crossref]

Chang, Y. C.

Chen, Y. C.

Chu, H.

De Martino, A.

Drévillon, B.

Emory, S. R.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc. 121(39), 9208–9214 (1999).
[Crossref]

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

Gaiduk, A. A.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Gaponenko, S. V.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Gaylord, T. K.

Gordon, J. P.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. 48(14), 957–960 (1982).
[Crossref]

Grann, E. B.

Hilfiker, J. N.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Hofmann, T.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

Hsu, S. H.

S. H. Hsu, Y. C. Chang, Y. C. Chen, P. K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18(2), 1310–1315 (2010).
[Crossref] [PubMed]

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5(5), 1194–1197 (2008).
[Crossref]

Hsu, S.-H.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Jiang, J.

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B 104(50), 11965–11971 (2000).
[Crossref]

Jupille, J.

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B 61(11), 7722–7733 (2000).
[Crossref]

Käll, M.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(33 Pt B), 4318–4324 (2000).
[Crossref] [PubMed]

Kaplan, B.

Kim, T. J.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Kim, Y. D.

S. H. Hsu, Y. C. Chang, Y. C. Chen, P. K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18(2), 1310–1315 (2010).
[Crossref] [PubMed]

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5(5), 1194–1197 (2008).
[Crossref]

Krug, J. T.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc. 121(39), 9208–9214 (1999).
[Crossref]

Kulakovich, O. S.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Kuo, H. C.

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
[Crossref]

Lazzari, R.

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films 419(1-2), 124–136 (2002).
[Crossref]

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B 61(11), 7722–7733 (2000).
[Crossref]

Li, G.

Liao, P. F.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. 48(14), 957–960 (1982).
[Crossref]

Lin, C. J.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Lin, G. R.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
[Crossref]

Lin, H. S.

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
[Crossref]

Liu, E. S.

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
[Crossref]

Liu, E.-S.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Maskevich, S. A.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Mewe, A.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

Michaels, A. M.

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B 104(50), 11965–11971 (2000).
[Crossref]

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Moharam, M. G.

Moirangthem, R. S.

Nie, S.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc. 121(39), 9208–9214 (1999).
[Crossref]

Nie, S. M.

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

Novikova, T.

Opsal, J.

Poelsema, B.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

Pommet, D. A.

Prokhorov, O. A.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Rekveld, S.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

Roux, S.

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B 61(11), 7722–7733 (2000).
[Crossref]

Sarangan, A.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

Schmidt, D.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

Schubert, E.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

Schubert, M.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

Shelekhina, V. M.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Simonsen, I.

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films 419(1-2), 124–136 (2002).
[Crossref]

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B 61(11), 7722–7733 (2000).
[Crossref]

Stefan Kooij, E.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

Strekal, N. D.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Videen, G.

Vlieger, J.

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

Wang, G. D.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc. 121(39), 9208–9214 (1999).
[Crossref]

Wei, P. K.

Wei, P.-K.

Wokaun, A.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. 48(14), 957–960 (1982).
[Crossref]

Wormeester, H.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

Xu, H.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(33 Pt B), 4318–4324 (2000).
[Crossref] [PubMed]

Yamamoto, G.

Ann. Phys. (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94(1), 011914 (2009).
[Crossref]

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett. 90(18), 181923 (2007).
[Crossref]

Biomed. Opt. Express (1)

Eur. Phys. J. B (1)

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B 24(2), 267–284 (2001).
[Crossref]

J. Am. Chem. Soc. (1)

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc. 121(39), 9208–9214 (1999).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Phys. Chem. B (1)

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B 104(50), 11965–11971 (2000).
[Crossref]

JETP Lett. (1)

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett. 74(6), 309–311 (2001).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B 61(11), 7722–7733 (2000).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(33 Pt B), 4318–4324 (2000).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. 48(14), 957–960 (1982).
[Crossref]

Phys. Status Solidi A (1)

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205(4), 876–879 (2008).
[Crossref]

Phys. Status Solidi C (1)

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5(5), 1194–1197 (2008).
[Crossref]

Science (1)

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

Thin Solid Films (2)

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[Crossref]

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films 419(1-2), 124–136 (2002).
[Crossref]

Other (5)

E. D. Palik, ed., Handbook of Optical Constants of Solids, vol. 1 (Academic, Orlando, FL, USA, 1985).

See for example, Fayyazuddin and Riazuddin, Quantum Mechanics (World Scientific, 1990), p. 368.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer Tracts in Modern Physics, Vol. 111 (Springer-Verlag, New York, 1988).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces (Imperial College Press, London, UK, 2002).

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Figures (6)

Fig. 1
Fig. 1

SEM pictures of samples with random distribution of Au nanoparticles with diameters: (a) 20nm, (b) 40nm, (c) 60nm, and (d) 80nm.

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Fig. 2
Fig. 2

(a) a picture which describes randomly distributed identical nanoparticles with variable distances between the centers of particles R i and the origin O (top view); (b) equivalent spheroid model to describe clusters with three different diameters (top view).

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Fig. 3
Fig. 3

The calculate field strength, | E | at the top of the sphere as a function of photon energy for light scattering from an isolated Au sphere obtained by the present Green’s function method with different cutoffs: ( ℓ max ,Nk) = (2,101) (dash-dotted), (4,51) (dotted), and (4,101) (dashed) and Mie scattering theory (solid). (a) d = 20 nm, (b) d = 40 nm, (c) d = 60 nm, and (d) d = 80 nm.

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Fig. 4
Fig. 4

SE measurements (solid curves) and model calculations (dash-dotted curves) without clusters for random distribution of nanoparticles with nominal sizes of (a) 20 nm, (b) 40 nm, (c) 60 nm, and (d) 80 nm for incident angles of 55°, 60°, and 65°.

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Fig. 5
Fig. 5

The ellipsometric parameters, Ψ and Δ as functions of photon energy obtained by the SHGF method for four different isolated pancakes of diameters (a) 2d, (b) 2.5d, (c) 3d and (d) 3.5d with d = 60nm at three different angles of incidence: 55° (solid line), 60 o (dashed line) and 65 o (dash-dotted line) on the substrate.

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Fig. 6
Fig. 6

SE measurements (solid curves) and model calculations (dashed dot curves) including clusters for random distribution of nanoparticles with nominal sizes of (a) 40, (b) 60, and (c) 80 nm for incident angles of 55°, 60°, and 65°.

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Tables (1)

Tables Icon

Table 1 Best-fit parameters used in the theoretical modeling for Au nanoparticles without and with clusters.

View Table

Equations (15)

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E( r )= 1 N ∑ i e i k 0 ⋅ R i u i ( r i ) ,
e i k 0 ⋅ R i u i ( r i )= N E 0 ( r )+ ∑ j=1 N ∫ d r ′ G( r, r ′ )⋅ V j ( r ′ ) e i k 0 ⋅ R j u j ( r ′ j ) ,
G( r, r ′ )= 1 ( 2π ) 2 ∫ d k n e i k n ⋅( ρ− ρ ′ ) g n ( z, z ′ ),
u i ( r i )= N e −i k 0 ⋅ R i E 0 ( r )+ ∑ j=1 N ∫ d ϕ n ( 2π ) 2 ∫ k n d k n ∫ d r ′ j e i k n ⋅( ρ i − ρ ′ j ) g n ( z, z ′ )⋅ e −i K n ⋅( R j − R i ) V j ( r ′ j ) u j ( r ′ j )
u i ( r i )= N e −i k 0 ⋅ R i E 0 ( r )+ ∫ d ϕ n ( 2π ) 2 ∫ k n d k n S( K n ) ∫ d r ′ i e i k n ⋅( ρ i − ρ ′ i ) g n ( z, z ′ )⋅ V 1 ( r ′ i ) u i ( r ′ i ) ,
S( K n )= 1 N ∑ i ∑ j e −i K n ⋅( R j − R i )
S( K n )=N ∑ G δ K n ,G ,
S( K n )=1+ f S 1 ( K n ),
S 1 ( K n )= 1 N ∑ i ∑ j≠i e −i K n ⋅( R j − R i ) .
S 1 ( K n )= 1 N ∑ i ∑ j≠i e −i K n ⋅( R j − R i ) ≈ ∑ j≠1 e −i K n ⋅( R j − R 1 ) ≈ 1 A cell ∫ dϕ ∫ R u ∞ e i K n ⋅R RdR = δ k n , k 0 −2 J 1 ( K n R u ) K n R u ,
S 1 ( K n )= ∑ j≠1 e −i K n ⋅( R j − R 1 ) e − ( R j − R 1 ) 2 / λ c 2 ≈ π λ c 2 A cell e − K n 2 λ c 2 /2 − 2π A cell ∫ 0 R u J 0 ( K n R u ) e − R 2 / λ c 2 RdR .
u 1 ( r )= N E 0 ( r )+ ∫ d ϕ n ( 2π ) 2 ∫ k n d k n ∫ d r ′ e i k n ⋅( ρ− ρ ′ ) g n ( z, z ′ )⋅{ [ 1+ S 1 ( K n ) f u f ] V 1 ( r ′ ) u 1 ( r ′ ) + ∑ α p α f S 1 ( K n ) V α ( r ′ ) c α ( r ′ ) },
c α ( r )= N E 0 ( r )+ ∫ d ϕ n ( 2π ) 2 ∫ k n d k n d r ′ e i k n ⋅( ρ− ρ ′ ) g n ( z, z ′ ) ⋅{ [ 1+ S 1 ( K n ) p α f ] V α ( r ′ ) c α ( r ′ ) + ∑ α ′ ≠α S 1 ( K n ) p α ′ f V α ′ ( r ′ ) c α ′ ( r ′ ) + f u f S 1 ( K n ) V 1 ( r ′ ) u 1 ( r ′ ) }
p( r )= N e −i k 0 ⋅ R i E 0 ( r )+ ∫ d ϕ n ( 2π ) 2 ∫ k n d k n [ 1+ S p ( K n )f ] ∫ d r ′ i e i k n ⋅( ρ− ρ ′ ) g n ( z, z ′ )⋅ V 1 ( r ′ )p( r ′ ) ,
E( z=0 )= E 0 ( z=0 ) + 1 A cell ∫ dz g 0 ( 0, z ′ )⋅ ∫ d ϕ ′ ∫ ρ ′ d ρ ′ e −i k 0 ⋅ ρ ′ [ f u V 1 ( r ′ ) u 1 ( r ′ )+ ∑ α p α V α ( r ′ ) c α ( r ′ ) + f p V 1 ( r ′ )p( r ′ )].

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