## Sensing properties of lattice resonances of 2D metal nanoparticle arrays: An analytical model |

Optics Express, Vol. 21, Issue 22, pp. 27490-27502 (2013)

http://dx.doi.org/10.1364/OE.21.027490

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### Abstract

Theoretical study of sensing properties of lattice resonances supported by arrays of gold nanoparticles expressed in terms of the figure of merit (FOM) is reported. Analytical expressions for the FOM for surface and bulk refractive index changes are derived to establish the relationship between the sensing performance and design parameters and to allow for the design of nanoparticle arrays with optimal sensing performance. It is demonstrated that lattice resonances exhibit about two orders of magnitude higher bulk FOM than localized surface plasmon (LSP) resonance and that the surface FOM provided by lattice resonances and LSP resonances are comparable.

© 2013 Optical Society of America

## 1. Introduction

1. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics **5**(2), 83–90 (2011). [CrossRef]

2. K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B. **3**(3), 430–440 (1986). [CrossRef]

3. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. **40**(11), 2281–2291 (1993). [CrossRef]

4. S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**(23), 10871–10875 (2004). [CrossRef] [PubMed]

5. S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. **121**(24), 12606–12612 (2004). [CrossRef] [PubMed]

6. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

7. B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B **82**(15), 155447 (2010). [CrossRef]

8. E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett. **5**(6), 1065–1070 (2005). [CrossRef] [PubMed]

9. Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. **93**(18), 181108 (2008). [CrossRef]

10. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. **101**(14), 143902 (2008). [CrossRef] [PubMed]

11. P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano **5**(6), 5151–5157 (2011). [CrossRef] [PubMed]

11. P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano **5**(6), 5151–5157 (2011). [CrossRef] [PubMed]

12. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. **108**(2), 462–493 (2008). [CrossRef] [PubMed]

## 2. Transmission spectrum of arrays on nanoparticles

*α*. When a dipole is excited by an electromagnetic wave, it reradiates a scattered wave with an amplitude proportional to its dipole moment. The field acting on a dipole is then a sum of the incident field plus the field radiated by all other dipoles. Assuming that the induced polarization for each NP is the same, an analytical expression for the effective polarizability is

*G*is the lattice sum, which accounts for the field scattered by the array [4

4. S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**(23), 10871–10875 (2004). [CrossRef] [PubMed]

*a,b,c*, and polarizability

*α*. In the electrostatic approximation

*L*is a shape factor (

*ε*and

*ε*are the dielectric constants of the NP and surrounding medium, respectively [13]. In these simulations, we assume that the NPs are made of gold (dielectric constant was taken from [14

_{m}14. P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

15. 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]

**R**

*are the positions of NPs [16*

_{n}16. F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**(4), 1267–1290 (2007). [CrossRef]

*Λ*is presented in Fig. 2.

*G*and

*T*under normal incidence can be expressed asFigure 3 shows the transmittance through the array as a function of wavelength calculated using Eq. (4). When the lattice sum reaches a maximum, the transmittance increases to one and at this wavelength the array of NPs becomes invisible. This behavior is referred to as Rayleigh anomaly (

*λ*). Since

_{LSP}*λ*. The narrow resonance is associated with so-called lattice resonances (

_{LSP}*λ*) and can be excited only when the Rayleigh anomaly occurs at longer wavelengths than LSP (

_{r}*δ*can be approximated byIn order to satisfy Eq. (5) for lattice resonances, two conditions must be fulfilled. The first condition,

*W*:

## 3. Sensitivity of the lattice resonance

*S*is defined as the sensitivity to changes in the refractive index of an infinite homogeneous medium surrounding the NP. The surface sensitivity

_{B}*S*is defined as the sensitivity to changes in the refractive index within a limited distance from the surface of the NP.

_{S}*λ*for a sufficient number of NPs, changes in the surface refractive index do not produce a shift of the Rayleigh anomaly. By differentiating the lattice resonance condition [Eq. (5)], the sensitivity to surface refractive index changes can be expressed asThe inverse polarizability of a core-shell ellipsoid NP with semiaxes

_{RA}*a, b, c,*and a shell thickness

*Δ*can be described by the following equation

*ε, ε*, and

_{m}*ε*are the permittivity of the NP, surrounding medium, and dielectric shell, respectively [13]. When the thickness of the shell is much smaller than the size of the NP,

_{s}## 4. Figure of merit

*A*and

*B*areUsing Eq. (3), for

*FOM*and

_{S}*FOM*on

_{B}*δ*for different resonant wavelengths, calculated using Eq. (15) and Eq. (16). The parameter

*δ*was assumed to lie within the range

*FOM*and

_{S}*FOM*exhibit a maximum at the optimized values of

_{B}*δ*where

*FOM*increases with the resonant wavelength and shape factor

_{B}*L*(which decreases with

*A*) and

*FOM*exhibits a maximum at a specific resonance wavelength that depends on the number and shape of the NPs.

_{S}*FOM*on the spectral distance between the Rayleigh anomaly and the lattice resonance was explored in an experimental study involving arrays of gold NPs with the radius of 47.5 – 82.5 nm, the height of 50 nm, and the periods of 300 – 600 nm [11

_{B}11. P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano **5**(6), 5151–5157 (2011). [CrossRef] [PubMed]

*υ*and

_{r}*υ*were the lattice resonance and Rayleigh anomaly frequencies [11

_{RA}**5**(6), 5151–5157 (2011). [CrossRef] [PubMed]

*FOM*on the radius of the NP for different shape factors and numbers of NPs. The period of the array was set to satisfy the resonance condition for a given wavelength according to Eq. (6). Both

*FOM*and

_{S}*FOM*exhibit a strong dependence on the radius of NP, having a single maximum at a specific radius. The values of the optimized radius of NP yielding the highest values of

_{B}*FOM*and

_{S}*FOM*correspond to the optimized value of parameter

_{B}*δ*and according to Eq. (6) can be expressed asAs apparent from Fig. 8, when the NP size increases above the optimum value, both the types of

*FOM*decrease (due to the broadening of the resonance feature) and the array starts to behave as an infinite array. Both

*FOM*values also decrease when the size of NPs is smaller than the optimum value. This decrease is associated with the fact that losses introduced by the finiteness of an array start to dominate and the width of the resonance feature increases. As the size of the NP continues to decrease, the resonance feature eventually disappears, where the condition

*FOM*increases while the

_{S}*FOM*decreases [Figs. 8(c) and 8(d)]. Further decrease in the shape factor results in disappearance of the lattice resonance, where the condition

_{B}*FOM*and

_{S}*FOM*decrease with decreasing number of particles [Figs. 8(a) and 8(b)].

_{B}*FOM*and

_{S}*FOM*can be improved by decreasing the size and increasing the number of NPs. In addition,

_{B}*FOM*can be improved by decreasing the size and shape factor. However, it should be noted that decreasing the size of the particles reduces the depth (contrast) of the resonance feature [Fig. 4(a)], rendering arrays of small particles rather impractical.

_{S}*FOM*using Eqs. (18) and (19) were compared with the exact values obtained by calculating the width of the transmission dip using Eq. (4), and furthermore, calculating the surface and bulk sensitivities by determining the shift of the resonance wavelength in response to a bulk refractive index change

*FOM*as a function of wavelength for different NP shape factors

*L*. The size of NP was optimized to achieve the maximum

*FOM*with at least 10% contrast of the resonance feature for each wavelength. The number of particles in the array was set to 2000 × 2000 to correspond with the area of a sensing substrate used typically in plasmonic sensing. The results suggest that both the surface and bulk refractive index

*FOM*can be maximized by choosing the optimum resonance wavelength. The optimal value of

*FOM*initially increases with wavelength and, after reaching a maximum, slowly decreases. While the initial growth of

_{S}*FOM*with wavelength is also rather rapid, it does not reach a local maximum and continues to increase albeit slowly with wavelength.

_{B}*FOM*for lattice resonance on ordered arrays of metal NPs were compared with the case of a LSP supported by a non-ordered array of non-interacting NPs (dots in Fig. 9). According to electrostatic theory,

*FOM*decreases with the size of NP and

_{S}*FOM*is size-independent [17

_{B}17. P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases **3**(3), FD4–FD11 (2008). [CrossRef] [PubMed]

*r =*10 nm (smaller NPs are challenging to fabricate). The comparison suggests that the lattice resonances on ordered arrays of metal NPs can provide

*FOM*which are larger by about two orders of magnitude than

_{B}*FOM*figures offered by LSP. On the other hand, the difference in the values for

_{B}*FOM*, which is the relevant quantity for most of the biosensing applications, seems to be rather minor.

_{S}## 5. Conclusion

## Appendix

**Q**[4

4. S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**(23), 10871–10875 (2004). [CrossRef] [PubMed]

**g**where the first term correspond to summation of (22) over

*G*represents the

_{0}*n = 0*term that can be expressed asWhen all diffracted waves other than the zero-th order are evanescent

16. F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**(4), 1267–1290 (2007). [CrossRef]

*N*, the sum in Eq. (22) can be reduced towhere the sinc function is derived from the restriction of the infinite sum by a rectangular function. For

*k*close to

*g*, the imaginary part can be written aswhere the sinc function was approximated by the Lorentz curve to eliminate the effect of fast oscillation. The condition for lattice resonance excitation can be satisfied for

_{1}## Acknowledgments

## References and links

1. | L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics |

2. | K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B. |

3. | V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. |

4. | S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. |

5. | S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. |

6. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

7. | B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B |

8. | E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett. |

9. | Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. |

10. | B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. |

11. | P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano |

12. | J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. |

13. | C. F. H. Bohren, D. R, |

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

15. | A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation damping in surface-enhanced Raman-scattering,” Phys. Rev. Lett. |

16. | F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. |

17. | P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(280.1415) Remote sensing and sensors : Biological sensing and sensors

(160.4236) Materials : Nanomaterials

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Plasmonics

**History**

Original Manuscript: September 5, 2013

Revised Manuscript: October 4, 2013

Manuscript Accepted: October 6, 2013

Published: November 4, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

Surface Plasmon Photonics (2013) *Optics Express*

**Citation**

Barbora Špačková and Jiří Homola, "Sensing properties of lattice resonances of 2D metal nanoparticle arrays: An analytical model," Opt. Express **21**, 27490-27502 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27490

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### References

- L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011). [CrossRef]
- K. T. Carron, W. Fluhr, M. Meier, A. Wokaun, and H. W. Lehmann, “Resonances of two-dimensional particle gratings in surface-enhanced Raman-scattering,” J. Opt. Soc. Am. B.3(3), 430–440 (1986). [CrossRef]
- V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt.40(11), 2281–2291 (1993). [CrossRef]
- S. L. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120(23), 10871–10875 (2004). [CrossRef] [PubMed]
- S. L. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121(24), 12606–12612 (2004). [CrossRef] [PubMed]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- B. Auguie, X. M. Bendana, W. L. Barnes, and F. J. G. de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B82(15), 155447 (2010). [CrossRef]
- E. M. Hicks, S. L. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett.5(6), 1065–1070 (2005). [CrossRef] [PubMed]
- Y. Z. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008). [CrossRef]
- B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101(14), 143902 (2008). [CrossRef] [PubMed]
- P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. C. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano5(6), 5151–5157 (2011). [CrossRef] [PubMed]
- J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev.108(2), 462–493 (2008). [CrossRef] [PubMed]
- C. F. H. Bohren, D. R, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).
- P. B. Johnson and R. W. Christy, “Optical-constants of noble-metals,” Phys. Rev. B6(12), 4370–4379 (1972). [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]
- F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys.79(4), 1267–1290 (2007). [CrossRef]
- P. Kvasnička and J. Homola, “Optical sensors based on spectroscopy of localized surface plasmons on metallic nanoparticles: Sensitivity considerations,” Biointerphases3(3), FD4–FD11 (2008). [CrossRef] [PubMed]

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