## Polarisation charges and scattering behaviour of realistically rounded plasmonic nanostructures |

Optics Express, Vol. 21, Issue 18, pp. 21500-21507 (2013)

http://dx.doi.org/10.1364/OE.21.021500

Acrobat PDF (1344 KB)

### Abstract

We study the effect of realistically rounding nanorod antennae and gap antennae on their far field and near field properties. The simulations show that both scattering behaviour and polarisation charge distribution depend significantly on rounding. Rounding is also seen to have a major effect on coupling between nanostructures. The results suggest that it is important to incorporate the effect of rounding to be able to design plasmonic nanostructures with desired properties.

© 2013 OSA

## 1. Introduction

2. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

3. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. **23**, 623–630 (1975). [CrossRef]

4. P. Monk, *Finite Element Methods for Maxwell’s Equations* (Oxford University, 2003). [CrossRef]

5. W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. **103**, 869–875 (1995). [CrossRef]

6. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E **58**, 3909–3915 (1998). [CrossRef]

7. F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B **65**, 115418 (2002). [CrossRef]

8. U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B **72**,195429 (2005). [CrossRef]

9. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. **26**, 732–740 (2009). [CrossRef]

10. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. **108**, 494–521 (2008). [CrossRef] [PubMed]

11. X. Lu, M. Rycenga, S. E. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. **60**, 167–192 (2009). [CrossRef]

12. A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett. **11**, 482–487 (2011). [CrossRef] [PubMed]

13. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science **308**, 1607–1609 (2005). [CrossRef] [PubMed]

14. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. **101**, 047401 (2008). [CrossRef] [PubMed]

16. N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. **8**, 758–762 (2009). [CrossRef]

17. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B **11**, 1732–1740 (1975). [CrossRef]

18. M. A. Yurkin and M. Kahnert, “Light scattering by a cube: Accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer **123**, 176–183 (2013). [CrossRef]

22. H. Chen, Z. Sun, W. Ni, K. C. Woo, H.-Q. Lin, L. Sun, C. Yan, and J. Wang, “Plasmon coupling in clusters composed of two-dimensionally ordered gold nanocubes,” Small **5**, 2111–2119 (2009). [CrossRef] [PubMed]

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

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

25. N. Grillet, D. Manchon, F. Bertorelle, C. Bonnet, M. Broyer, E. Cottancin, J. Lermé, M. Hillenkamp, and M. Pellarin, “Plasmon coupling in silver nanocube dimers: Resonance splitting induced by edge rounding,” ACS Nano **5**, 9450–9462 (2011). [CrossRef] [PubMed]

26. M. B. Cortie, F. Liu, M. D. Arnold, and Y. Niidome, “Multimode resonances in silver nanocuboids,” Langmuir **28**, 9103–9112 (2012). [CrossRef] [PubMed]

## 2. Formulation of the problem

*l*,

*b*,

*h*. Suppose we need to round the edges and corners with a rounding radius of

*r*such that 2

*r*≤ min(

*l*,

*b*,

*h*). This can be done ensuring continuity and differentiability of the surfaces by making a composite structure consisting of seven cuboids, twelve quarter-cylinders and eight sphere-octants. The assembly of a rounded cuboid from these pieces is shown in Fig. 1. It is easy to verify that all boundaries between meeting surfaces are smooth.

*YZ*plane cross section of the cuboids was kept constant as

*b*=

*h*= 40 nm while the length

*l*along the

*X*dimension of the structures and the radius of rounding

*r*were varied. In the case of the gap antenna, two such identical cuboids were placed symmetrically such that their

*YZ*-plane surfaces faced each other. In all the simulations presented here, a plane wave propagating in the

*Z*direction and polarised along

*X*was used for illuminating the structures. The nanostructures being simulated are made out of silver, unless explicitly mentioned otherwise. The dielectric function for silver was taken from Johnson and Christy [27

27. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

9. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. **26**, 732–740 (2009). [CrossRef]

^{2}for all simulations.

*μm*, and normalising the result to the incident field intensity: For this, scattered electric and magnetic fields were calculated at 1050 nearly equidistant points on the sphere [28

28. A. M. Kern and O. J. F. Martin, “Pitfalls in the determination of optical cross sections from surface integral equation simulations,” IEEE Trans. Antennas Propag. **58**, 2158–2161 (2010). [CrossRef]

## 3. Results and discussion

9. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. **26**, 732–740 (2009). [CrossRef]

12. A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett. **11**, 482–487 (2011). [CrossRef] [PubMed]

28. A. M. Kern and O. J. F. Martin, “Pitfalls in the determination of optical cross sections from surface integral equation simulations,” IEEE Trans. Antennas Propag. **58**, 2158–2161 (2010). [CrossRef]

17. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B **11**, 1732–1740 (1975). [CrossRef]

26. M. B. Cortie, F. Liu, M. D. Arnold, and Y. Niidome, “Multimode resonances in silver nanocuboids,” Langmuir **28**, 9103–9112 (2012). [CrossRef] [PubMed]

29. S. Zhang, K. Bao, N. J. Halas, H. Xu, and P. Norlander, “Substrate-induced fano resonances of a plasmonic nanocube: a route to increased-sensitivity localized surface plasmon resonance sensors revealed,” Nano Lett. **11**, 1657–1663 (2011). [CrossRef] [PubMed]

27. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*r*= 3 nm) and that for an arm of a gap antenna formed by two such nanorods separated by 10 nm (

*r*= 3 nm and

*r*= 9 nm) are plotted in Fig. 7. As expected, for the single cuboid, the polarisation charges are concentrated at the corners. However, for the gap antenna, the charges are seen to be spread all over the surface facing the gap. This can be explained by the fact that there is an opposite charge at the matching face of the other arm of the gap antenna, attracting charge on this arm towards it. The two oppositely charged faces with charges almost uniformly spread over them acts as a capacitor, providing high field coupling between the faces. However, when the rounding of the cuboids is increased, the situation changes. The polarisation charges are still attracted towards the gap face, but are now more dispersed around the curved edges. It is important to note that the flat part of the face is what is closest to the other antenna arm, thus providing maximum coupling. The presence of charges away from the flat region on the edges reduces the coupling.

*δ*. Now consider making a gap antenna out of two such identical ideal cuboids separated by

_{s}*d*. Due to the field coupling between the two arms, the peak scattering wavelength for the ideal gap antenna is red shifted with respect to the peak for the ideal rod antenna by a value

*d*is red shifted with respect to the peak for the rounded rod antenna by a value

*δ*>

_{g}*δ*. That is, the rounding induced blue shift is higher for a gap antenna as compared to a single cuboid antenna.

_{s}## 4. Conclusion

30. A. Unger and M. Kreiter, “Analysing the performance of plasmonic resonators for dielectric sensing,” J. Phys. Chem. C **113**, 12243–12251 (2009). [CrossRef]

31. A. Lovera, B. Gallinet, P. Norlander, and O. J. F. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano **7**, 4527–4536 (2013). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. A. Maier, |

2. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

3. | A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. |

4. | P. Monk, |

5. | W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. |

6. | O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E |

7. | F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B |

8. | U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B |

9. | A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. |

10. | M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. |

11. | X. Lu, M. Rycenga, S. E. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. |

12. | A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett. |

13. | P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science |

14. | S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. |

15. | N. Verellen, Y. Sonnefraud, H. Sobhani, V. V. Moshchalkov, P. Van Dorpe, P. Norlander, and S. A. Maier, “Fano resonances in coherent plasmonic nanocavities,” Nano Lett. |

16. | N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. |

17. | R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B |

18. | M. A. Yurkin and M. Kahnert, “Light scattering by a cube: Accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer |

19. | W. J. Galush, S. A. Shelby, M. J. Mulvihill, A. Tao, P. Yang, and J. T. Groves, “A nanocube plasmonic sensor for molecular binding on membrane surfaces,” Nano Lett. |

20. | L. J. Sherry, S.-H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and L. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. |

21. | M. Rycenga, J. M. McLellan, and Y. Xia, “Controlling the assembly of silver nanocubes through selective functionalization of their faces,” Adv. Mater. |

22. | H. Chen, Z. Sun, W. Ni, K. C. Woo, H.-Q. Lin, L. Sun, C. Yan, and J. Wang, “Plasmon coupling in clusters composed of two-dimensionally ordered gold nanocubes,” Small |

23. | W. Li, P. H. C. Camargo, X. Lu, and Y. Xia, “Dimers of silver nanospheres: facile synthesis and their use as hot spots for surface-enhanced Raman scattering,” Nano Lett. |

24. | M. Rycenga, C. M. Cobley, J. Zeng, W. Li, C. H. Moran, Q. Zhang, D. Qin, and Y. Xia, “Controlling the synthesis and assembly of silver nanostructures for plasmonic applications,” Chem. Rev. |

25. | N. Grillet, D. Manchon, F. Bertorelle, C. Bonnet, M. Broyer, E. Cottancin, J. Lermé, M. Hillenkamp, and M. Pellarin, “Plasmon coupling in silver nanocube dimers: Resonance splitting induced by edge rounding,” ACS Nano |

26. | M. B. Cortie, F. Liu, M. D. Arnold, and Y. Niidome, “Multimode resonances in silver nanocuboids,” Langmuir |

27. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

28. | A. M. Kern and O. J. F. Martin, “Pitfalls in the determination of optical cross sections from surface integral equation simulations,” IEEE Trans. Antennas Propag. |

29. | S. Zhang, K. Bao, N. J. Halas, H. Xu, and P. Norlander, “Substrate-induced fano resonances of a plasmonic nanocube: a route to increased-sensitivity localized surface plasmon resonance sensors revealed,” Nano Lett. |

30. | A. Unger and M. Kreiter, “Analysing the performance of plasmonic resonators for dielectric sensing,” J. Phys. Chem. C |

31. | A. Lovera, B. Gallinet, P. Norlander, and O. J. F. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 9, 2013

Revised Manuscript: August 27, 2013

Manuscript Accepted: August 29, 2013

Published: September 5, 2013

**Citation**

T. V. Raziman and Olivier J. F. Martin, "Polarisation charges and scattering behaviour of realistically rounded plasmonic nanostructures," Opt. Express **21**, 21500-21507 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21500

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### References

- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966). [CrossRef]
- A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech.23, 623–630 (1975). [CrossRef]
- P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University, 2003). [CrossRef]
- W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys.103, 869–875 (1995). [CrossRef]
- O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E58, 3909–3915 (1998). [CrossRef]
- F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B65, 115418 (2002). [CrossRef]
- U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B72,195429 (2005). [CrossRef]
- A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am.26, 732–740 (2009). [CrossRef]
- M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev.108, 494–521 (2008). [CrossRef] [PubMed]
- X. Lu, M. Rycenga, S. E. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009). [CrossRef]
- A. M. Kern and O. J. F. Martin, “Excitation and reemission of molecules near realistic plasmonic nanostructures,” Nano Lett.11, 482–487 (2011). [CrossRef] [PubMed]
- P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308, 1607–1609 (2005). [CrossRef] [PubMed]
- S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett.101, 047401 (2008). [CrossRef] [PubMed]
- N. Verellen, Y. Sonnefraud, H. Sobhani, V. V. Moshchalkov, P. Van Dorpe, P. Norlander, and S. A. Maier, “Fano resonances in coherent plasmonic nanocavities,” Nano Lett.9, 1663–1667 (2009). [CrossRef] [PubMed]
- N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009). [CrossRef]
- R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B11, 1732–1740 (1975). [CrossRef]
- M. A. Yurkin and M. Kahnert, “Light scattering by a cube: Accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer123, 176–183 (2013). [CrossRef]
- W. J. Galush, S. A. Shelby, M. J. Mulvihill, A. Tao, P. Yang, and J. T. Groves, “A nanocube plasmonic sensor for molecular binding on membrane surfaces,” Nano Lett.9, 2077–2082 (2009).
- L. J. Sherry, S.-H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and L. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett.5, 2034–2038 (2005). [CrossRef] [PubMed]
- M. Rycenga, J. M. McLellan, and Y. Xia, “Controlling the assembly of silver nanocubes through selective functionalization of their faces,” Adv. Mater.20, 2416–2420 (2008). [CrossRef]
- H. Chen, Z. Sun, W. Ni, K. C. Woo, H.-Q. Lin, L. Sun, C. Yan, and J. Wang, “Plasmon coupling in clusters composed of two-dimensionally ordered gold nanocubes,” Small5, 2111–2119 (2009). [CrossRef] [PubMed]
- W. Li, P. H. C. Camargo, X. Lu, and Y. Xia, “Dimers of silver nanospheres: facile synthesis and their use as hot spots for surface-enhanced Raman scattering,” Nano Lett.9, 485–490 (2009).
- M. Rycenga, C. M. Cobley, J. Zeng, W. Li, C. H. Moran, Q. Zhang, D. Qin, and Y. Xia, “Controlling the synthesis and assembly of silver nanostructures for plasmonic applications,” Chem. Rev.111, 3669–3712 (2011).
- N. Grillet, D. Manchon, F. Bertorelle, C. Bonnet, M. Broyer, E. Cottancin, J. Lermé, M. Hillenkamp, and M. Pellarin, “Plasmon coupling in silver nanocube dimers: Resonance splitting induced by edge rounding,” ACS Nano5, 9450–9462 (2011). [CrossRef] [PubMed]
- M. B. Cortie, F. Liu, M. D. Arnold, and Y. Niidome, “Multimode resonances in silver nanocuboids,” Langmuir28, 9103–9112 (2012). [CrossRef] [PubMed]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- A. M. Kern and O. J. F. Martin, “Pitfalls in the determination of optical cross sections from surface integral equation simulations,” IEEE Trans. Antennas Propag.58, 2158–2161 (2010). [CrossRef]
- S. Zhang, K. Bao, N. J. Halas, H. Xu, and P. Norlander, “Substrate-induced fano resonances of a plasmonic nanocube: a route to increased-sensitivity localized surface plasmon resonance sensors revealed,” Nano Lett.11, 1657–1663 (2011). [CrossRef] [PubMed]
- A. Unger and M. Kreiter, “Analysing the performance of plasmonic resonators for dielectric sensing,” J. Phys. Chem. C113, 12243–12251 (2009). [CrossRef]
- A. Lovera, B. Gallinet, P. Norlander, and O. J. F. Martin, “Mechanisms of Fano resonances in coupled plasmonic systems,” ACS Nano7, 4527–4536 (2013). [CrossRef] [PubMed]

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