## Effect of clustering on ellipsometric spectra of randomly distributed gold nanoparticles on a substrate |

Optics Express, Vol. 21, Issue 3, pp. 3091-3102 (2013)

http://dx.doi.org/10.1364/OE.21.003091

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### 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 OSA

## 1. Introduction

2. 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]

7. 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]

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). [CrossRef] [PubMed]

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). [CrossRef]

14. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. **14**(1), 29–49 (1975). [PubMed]

15. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A **8**(3), 483–489 (1991). [CrossRef]

16. 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]

19. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**(5), 1068–1076 (1995). [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]

21. Y. C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A **23**(3), 638–645 (2006). [CrossRef] [PubMed]

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]

21. Y. C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A **23**(3), 638–645 (2006). [CrossRef] [PubMed]

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]

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]

25. R. S. Moirangthem, Y. C. Chang, and P.-K. Wei, “Investigation of surface plasmon biosensing using gold nanoparticles enhanced ellipsometry,” Opt. Lett. **36**(5), 775–777 (2011). [CrossRef] [PubMed]

26. R. S. Moirangthem, Y. C. Chang, and P. K. Wei, “Ellipsometry study on gold-nanoparticle-coated gold thin film for biosensing application,” Biomed. Opt. Express **2**(9), 2569–2576 (2011). [CrossRef] [PubMed]

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]

## 2. Modeling for light scattering from randomly distributed nanoparticles with clusters

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]

**18**(2), 1310–1315 (2010). [CrossRef] [PubMed]

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]

**18**(2), 1310–1315 (2010). [CrossRef] [PubMed]

*p*(the pitch). For convenience, we can write the wave function for the electric field in the form of linear combination of localized orbitals (LCAO) (for

*z*axis,where

21. Y. C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A **23**(3), 638–645 (2006). [CrossRef] [PubMed]

**k**

*denotes any wave vector in the x-y plane, since the system is not periodic, unlike in Ref [21*

_{n}**23**(3), 638–645 (2006). [CrossRef] [PubMed]

**23**(3), 638–645 (2006). [CrossRef] [PubMed]

*f*, which describes the ratio of the average of wave functions over all other sites to the on-site wave function. If wave functions at all sites are identical and the coherence is maintained, we will have

*ε*(here for Au) if we choose

_{a}*ε*is a complex number, so is

_{a}*k*

_{1}. The basis set

**5**(5), 1194–1197 (2008). [CrossRef]

**18**(2), 1310–1315 (2010). [CrossRef] [PubMed]

**5**(5), 1194–1197 (2008). [CrossRef]

**18**(2), 1310–1315 (2010). [CrossRef] [PubMed]

*α*and vanishes otherwise.

*α*with

*α*. To avoid using too many fitting parameters, we assume that

*n*

_{c}is the number of different sizes of clusters included. Note that

*α*satisfies a similar L-S equation

*α*. The last term describes the coupling of the cluster to surrounding nanoparticles, which is non-negligible since

*f*is close to 1. For simplicity, we can approximate the average of all clusters of type

*α*by a spheroid with height

*h*and diameter

*d*, which occupies the volume covered by a circular revolution of a dimer. Similarly, the angular average of closed-packed clusters made of five to seven nanoparticles may be simulated by a pancake with diameter 3

*d*, which occupies the volume covered by a circular revolution of a chain of three nanoparticles, and the angular average of clusters made of eight to twelve nanoparticles is simulated by a pancake with diameter

*d*and 4

*d*. Thus, a model including pancakes with diameters evenly distributed between 2

*d*and

*2R*(the cut-off diameter) with suitable proportions

_{u}*R*replaced by

_{u}**E**-field on a plane just above the nanoparticles (taken to be at

*s*-polarized) and TM (

*p*-polarized) reflectivities are given by

## 3. Results and discussions

_{k}) = (2,101), (4,51) and (4,101), respectively. The numbers of

*z*and

_{k}) are needed to give convergent results. For (

_{k}) = (4,101), we find convergent results (dashed lines) for all cases, and they are in excellent agreement with the Mie theory, indicating the accuracy of our numerical implementation.

*z*meshes for the pancake like clusters due to their low aspect ratio of height (

*d*(diameter) and

*h*(height) of nanoparticles, the average distance between nanoparticles (pitch),

*f*. For the model including coupling with clusters, we adopt the best-fit parameters from the model without clusters and add two fitting parameters: the total fraction of area occupied by small clusters

*f*

_{c}and that for the patches of aggregated nanoparticles,

*f*

_{p}. Seven small clusters (

*n*

_{c}= 7) with diameters evenly distributed between 2

*d*and 2

*R*are used, and their heights are kept the same as the nanoparticles. We find that the results remain nearly unchanged even if we increase the sampling number of sizes for small clusters.

_{u}**18**(2), 1310–1315 (2010). [CrossRef] [PubMed]

*d*, 2.5

*d*, 3

*d*, and 3.5

*d*with

*d*= 60 nm at three different angles of incidence: 55° (solid lines), 60° (dashed lines) and 65° (dash-dotted lines). The number of

*z*meshes used in the integration is kept at 50 in all cases and the pitch (

*p*) used is 170 nm. In order to ensure the convergent results, the cutoff of angular momentum quantum number

_{ρ}), and the number of

_{k}) used in the integration for pancakes with lateral sizes 2

*d*, 2.5

*d*, 3

*d*, and 3.5

*d*are (

_{ρ,}N

_{k}) = (6,70,71), (6,80,81), (6,90,91), and (7,100,91), respectively. As can be seen in Fig. 5, these clusters represented by pancakes lead to larger enhancement and multiple plasmonic resonances at energies near 2 eV due to their enlarged cross-section in the x-y plane.

*f*

_{c}and

*f*

_{p}used to obtain the best fit are listed in Columns 5 and 6 of Table 1. A variation of

*f*

_{c}and

*f*

_{p}by more than

*f*is rescaled such that

*f*is the same as the

_{u}f*f*listed in Column 3 of Table 1. Including the clustering effect, we can obtain much better agreement with the experimental data for samples with nominal sizes of 40nm, 60nm, and 80nm, where there is obvious cluster formation as shown in Fig. 1(b)-1(d). The mean-squared errors (MSE’s) as shown in Column 6 of Table 1 now become much smaller. Examining the contributions, we found that the coupling with smaller clusters are responsible for the improvement for photon energies below 2.5 eV, where the plasmonic resonance plays an important role. Comparing the spectra in Fig. 6 and Fig. 5 for photon energies below 2.5 eV, we notice a qualitative difference in the spectral lineshapes with and without the clustering effect. Without the clustering effect, the

## 4. Conclusion

*p*), the fraction of areas occupied by small clusters (

*f*

_{p}). Such information is very useful for nondestructive metrology of nanoparitcles covered samples .

## Acknowledgments

## References and links

1. | H. Raether, |

2. | 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. |

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 |

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 |

5. | A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett. |

6. | S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science |

7. | 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. |

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. |

9. | H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films |

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 |

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. |

12. | G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. |

13. | C. F. Bohren and D. R. Huffman, |

14. | S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. |

15. | G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A |

16. | I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B |

17. | 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 |

18. | D. Bedeaux and J. Vlieger, |

19. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

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 |

21. | Y. C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A |

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 |

23. | R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films |

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. |

25. | R. S. Moirangthem, Y. C. Chang, and P.-K. Wei, “Investigation of surface plasmon biosensing using gold nanoparticles enhanced ellipsometry,” Opt. Lett. |

26. | R. S. Moirangthem, Y. C. Chang, and P. K. Wei, “Ellipsometry study on gold-nanoparticle-coated gold thin film for biosensing application,” Biomed. Opt. Express |

27. | E. D. Palik, ed., |

28. | See for example, Fayyazuddin and Riazuddin, |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(290.5820) Scattering : Scattering measurements

(290.5825) Scattering : Scattering theory

(240.2130) Optics at surfaces : Ellipsometry and polarimetry

**ToC Category:**

Scattering

**History**

Original Manuscript: December 6, 2012

Revised Manuscript: January 4, 2013

Manuscript Accepted: January 7, 2013

Published: January 31, 2013

**Virtual Issues**

Vol. 8, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Huai-Yi Xie, Yia-Chung Chang, Guangwei Li, and Shih-Hsin Hsu, "Effect of clustering on ellipsometric spectra of randomly distributed gold nanoparticles on a substrate," Opt. Express **21**, 3091-3102 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-3-3091

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### References

- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer Tracts in Modern Physics, Vol. 111 (Springer-Verlag, New York, 1988).
- 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]
- 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. Topics62(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. B104(50), 11965–11971 (2000). [CrossRef]
- 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]
- S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997). [CrossRef] [PubMed]
- 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]
- 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 Films455–456, 323–334 (2004). [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 A205(4), 876–879 (2008). [CrossRef]
- 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. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.330(3), 377–445 (1908). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt.14(1), 29–49 (1975). [PubMed]
- G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A8(3), 483–489 (1991). [CrossRef]
- I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B61(11), 7722–7733 (2000). [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. B24(2), 267–284 (2001). [CrossRef]
- D. Bedeaux and J. Vlieger, Optical Properties of Surfaces (Imperial College Press, London, UK, 2002).
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A12(5), 1068–1076 (1995). [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 C5(5), 1194–1197 (2008). [CrossRef]
- Y. C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films,” J. Opt. Soc. Am. A23(3), 638–645 (2006). [CrossRef] [PubMed]
- 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. Express18(2), 1310–1315 (2010). [CrossRef] [PubMed]
- R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films419(1-2), 124–136 (2002). [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]
- R. S. Moirangthem, Y. C. Chang, and P.-K. Wei, “Investigation of surface plasmon biosensing using gold nanoparticles enhanced ellipsometry,” Opt. Lett.36(5), 775–777 (2011). [CrossRef] [PubMed]
- R. S. Moirangthem, Y. C. Chang, and P. K. Wei, “Ellipsometry study on gold-nanoparticle-coated gold thin film for biosensing application,” Biomed. Opt. Express2(9), 2569–2576 (2011). [CrossRef] [PubMed]
- 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.

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