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Optics Express

Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15312–15324
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Light scattering from 2D arrays of monodispersed Ag-nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling

Sajal Biring, Huai-Hsien Wang, Juen-Kai Wang, and Yuh-Lin Wang  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15312-15324 (2008)
http://dx.doi.org/10.1364/OE.16.015312


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Abstract

Two dimensional arrays of monodispersed Ag-nanoparticles separated by different gaps with sub-10 nm precision are fabricated on anodic alumina substrates with self-organized pores. Light scattering spectra from the arrays evolve with the gaps, revealing plasmonic coupling among the nanoparticles, which can be satisfactorily interpreted by analytical formulae derived from generic dipolar approximation. The general formulism lays down a foundation for predicting the Q factor of an array of metallic nano-particles and its geometric characteristics.

© 2008 Optical Society of America

1. Introduction

The electromagnetic interaction between nanometer-sized particles has been studied experimentally in the case of isolated pairs [10

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. 3, 1087–1090 (2003). [CrossRef]

-12

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]

] and the arrays [13

13. C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” J. Phys. Chem. B 107, 7337–7342 (2003).

] of particles fabricated by electron-beam lithography or nanosphere lithography. Through far-field spectroscopic measurements, the observed relationship between scattering spectrum and interparticle spacing was explained by dipole-dipole interaction between the particles [14

14. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003). [CrossRef]

]. This relationship was also investigated by numerical analysis like discrete dipole approximation [10

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. 3, 1087–1090 (2003). [CrossRef]

] and finite-difference time-domain method [15

15. J. P. Kotmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express 8, 655–663 (2001). [CrossRef]

]. However, only qualitative understanding was provided and simple analytical model is still missing, limiting our ability to design arrays of nanoparticles for technological applications. In addition, the experimental studies are limited to large disk-shaped nanoparticles made by electron-beam lithography or complex sculpted structures made by nanosphere lithography and other special deposition techniques [16

16. X. -D. Xiang, et al., “A Combinatorial Approach to Materials Discovery,” Science 268, 1738–1740 (1995). [CrossRef] [PubMed]

]. The complex geometry of these nanoparticles renders the interpretation of the spectra difficult. There have been several theoretical efforts in the establishment of simple comprehensive models. Persson and Liebsch studied the optical properties of two-dimensional metallic spherical particle arrays and derived an analytical formula for the dependence of the spectral peak on the interparticle spacing [17

17. B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).

]. In a study of scattering of light from one dimensional dipole chain, Markel derived an analytical formula for effective polarizability in long-wavelength limit and pointed out the connection between spectral characteristics and electromagnetic interaction [18

18. V. A. Markel, “Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure,” J. Mod. Opt. 40, 2281–2291 (1993). [CrossRef]

].

In this study, we investigate inter-particle interaction of arrays of Ag nanorods embedded in AAO nanochannel arrays using light-scattering spectroscopy. Using this system for the study has several advantages. Firstly, well-ordered nanochannels with tunable inter-channel spacings and diameters have been demonstrated with sub-10 nanometer precision [8

8. H. H. Wang, et al., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. 18, 491–495 (2006). [CrossRef]

], enabling the precise fabrication of an array of uniform Ag nanorods with desired size and spatial arrangement. Secondly, with controlled electrodeposition process, Ag nano-rods with smooth geometry can be produced, facilitating simple modeling of the optical properties of these nanorods. Thirdly, these Ag nanorods are surrounded mostly by alumina, which simplifies the effect from the surrounding medium. This is in great contrast to the other systems fabricated by electron-beam lithography and nanosphere lithography. These advantages therefore allow us to derive simple quantitative model to interpret the three spectroscopic characteristics (peak position, width and intensity) acquired from far-field scattering spectra.

Fig. 1. Variation in interparticle spacing (Δr) vs. mean interparticle spacing (r̄). Inset shows the histogram of interparticle spacing (r) for a sample with r̄=30 nm.

2. Sample preparation and measurements

Preparation of silver nanorods inside AAO has been described elsewhere [8

8. H. H. Wang, et al., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. 18, 491–495 (2006). [CrossRef]

]. Briefly, self-organized, hexagonally close-packed AAO nanochannels were fabricated by anodizing an electropolished aluminum foil. The spacing between nanochannels, r, was adjusted by varying the anodization voltage. A fixed diameter of 25 nm was prepared by subsequent etching of the channel walls after the anodization process. Finally, Ag nanorods were grown inside the channels by AC electro-deposition. In the spectroscopy study, the Ag/AAO sample was illuminated in dark-field mode by unpolarized white light from a halogen lamp in an inverted microscope. In dark-field configuration, the sample was illuminated obliquely and the scattered light was collected along the normal to the sample surface. It was then directed to a monochrometer plus charge-coupled device for spectral detection. The smallest mean interparticle spacing, r̄, achieved is 30 nm, yielding a mean gap of 5 nm [19

19. see Appendix.

]. The corresponding histogram of the interparticle spacing, as given in the inset of Fig. 1, shows a full-width at half-maximum, Δr, of 3 nm. Figure 1 displays Δr as a function of r̄ for all the samples prepared for the spectroscopic measurement. A typical scattering spectrum exhibits two peaks (not shown), corresponding to the transverse and longitudinal modes [20

20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York,1983), pp. 130.

] of the plasmonic resonance which can be assigned by polarization dependency of these two modes. The rod length of all the samples used in this study was adjusted to be approximately 100 nm such that only the transverse mode is visible in the spectral range of interest.

3. Theory

The Ag nanoparticle arrays in our study can be considered as two-dimensional hexagonal arrays made of Ag prolate spheroids in AAO matrix. In the case of arrays of particles, the local field Eloc(ri) for the particle located at ri is the sum of the incident field Einc(ri) and the fields induced by the rest of particles. The effective polarizability of the arrays can be derived accordingly:

αeff=αT1αTU,
(1)

where

U=Σji[k2sin2θijrijik(3cos2θij1)rij2+(3cos2θij1)rij3]eikrij,
(2)

αT is the polarizability of a single prolate spheroid of the transverse mode in an electrostatic field, k=2π/λ, θ ij is the angle between rij and Einc(ri), and rij=ri-rj. Considering long-wavelength approximation, Eq. (2) becomes

UF·r3+iB
(3)

where r is the center-to-center distance between two adjacent particles and F is the lattice sum of the hexagonal particle array. B=Σji(3cos2θij1)[sin(krij)r3ijkcos(krij)rij2], neglecting the far-field term. It can be shown that the resonance frequency, ΩT, the corresponding width, Γ, and the peak scattering intensity at ΩT, IT), are expressed as:

ΩT2=ωp2[1F(Sr)3](1+Cεm)+(εm1)F(Sr)3,
(4)
Γ=1τ+S3Bωp2(1+Cεm)ΩT+εm11+CεmS3BΩT,
(5)

and

I(ΩT)N(r)ΩT4αeff(ΩT)2=N(r){ΩTS3[(εm1)ΩT2+ωp2]Γ(1+Cεm)}2,
(6)

where εm is the dielectric constant of the surrounding medium, S 3=R 2 h/3L, C=(1-L)/L, R and 2h are the radius and the length of the spheroid, respectively, L is the depolarization factor, and N(r) is the surface particle density. In the derivation, Drude model is assumed for the dielectric function of Ag with plasma frequency, ωp, and relaxation time, τ, as parameters. The detailed derivation of Eqs. (4)-(6) is presented in Ref. 19. In the case of arrays of spherical particles in vacuum, the resultant resonance frequency agrees with the derived result by Persson and Liebsch [17

17. B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).

] and exhibits a dependence on R/r which was also obtained by many research groups [10

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. 3, 1087–1090 (2003). [CrossRef]

-12

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]

]. Equation (5) shows that the spectral width is also dependent on the coupling between adjacent particles in the array, which was ignored previously [10

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. 3, 1087–1090 (2003). [CrossRef]

-12

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]

, 17

17. B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).

]. As described above, the coupling by the adjacent dipoles introduces the imaginary term in U and therefore extra contributions to the imaginary part of αeff, as shown in Eq. (1). This additional quadrature component in αeff causes effective absorption that reflects the energy flow from a specific dipole to its surrounding dipoles and therefore gives rise to extra broadening in Γ. For narrower interparticle spacings, ΩT exhibits red shifting and B becomes larger, thus leading to an increase in Γ. Finally, Eq. (6) presents the fact that the peak intensity is proportional to the square of the Q factor of the system, which is equal to ΩT/Γ and directly reflects the field strength at the resonance frequency in this plasmonic system [21

21. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 371.

].

Fig. 2. Normalized scattering spectra from Ag nanoparticle arrays with mean interparticle spacings: 50 nm (violet line), 45 nm (blue line), 40 nm (green line), 35 nm (orange line), and 30 nm (red line).

4. Results and discussion

The physical origin of the observed spectral peak shifting and width broadening induced by plasmonic coupling can be understood using the theoretical framework as described above. There are three spatial regions of interest for the field generated from a single induced-dipole: near, intermediate and far zone [22

22. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1999), pp. 407.

], which are reflected by the three terms in Eq. (2). The 1/r 3 ij and 1/rij terms represent the near- and far-zone contributions, respectively, while the 1/r 2 ij term dominates in the intermediate zone and has a 90° phase shift with respect to the other terms. The far-zone contribution becomes prominent only when rij is comparable to the wavelength and therefore is insignificant in the discussion of the coupling induced peak shifting and broadening. Because the near-zone term is much larger than the intermediate-zone term under the long-wave approximation, it dominates the real part of U in the spatial range of this study, and acts as the single source in the coupling-induced peak shifting. In contrast, both the near- and intermediate-zone terms contribute to the spectral width broadening [18

18. V. A. Markel, “Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure,” J. Mod. Opt. 40, 2281–2291 (1993). [CrossRef]

]. Since the broadening is caused by the collective energy transfer between dipoles of different separations in the intermediate zone, this effect could not have been revealed by the studies of isolated dipole pairs reported previously [10

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. 3, 1087–1090 (2003). [CrossRef]

-12

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]

]. Such theoretical understanding allows us to conclude that the spectral peak shifting and width broadening are measures of the plasmonic coupling within the near and intermediate zones.

Fig. 3. Transverse-mode resonance peak (ΩT) and Lorentzian width (ΓL) vs. mean interparticle spacing (r̄). Red solid lines are theoretical fittings.

The spectral peak-shift analysis provides a foundation for subsequent analysis of the line-width broadening. In particular, the dependence of the spectral peak shifting on the interparticle spacing provides a means to incorporate the influence of the variation in the interparticle spacing, presented in Fig. 1, in the analysis of the spectral width broadening data. This bears the resemblance with the inhomogeneous contribution in atomic and molecular spectroscopy. According to Eq. (4), the dependence of the variation in ΩT, ΔΩT, on the variation in r, Δr, can be derived [19

19. see Appendix.

]. This formula was used to calculate the inhomogeneous contribution, which has a Gaussian distribution, to the observed spectral width based on the result in Fig. 1 and the extracted parameters from the spectral peak-shift analysis. On the basis of the relationship between the extracted Voigt width, ΓV, and r̄ [19

19. see Appendix.

], the Lorentzian width, ΓL, was then obtained according to the empirical relation among the Voigt, Gaussian and Lorentzian width [25

25. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt linewidth: A brief review” J. Quant. Spectrosc. Radiat. Transfer 17, 233–236 (1977). [CrossRef]

]. The dependence of the resultant Lorentzian width on r̄ is displayed in Fig. 3. ΓL was then fitted with Eq. (5) to obtain the parameters listed in the fourth column of Table 1. Similar to the spectral peak-shifting analysis, both ωp and τ obtained from the fitting are in good agreement with that extracted from fitting the optical constant of Ag to the Drude model.

Fig. 4. Scattering intensity at the peak of transverse-mode resonance, I *T), (filled circles) and Q factor (open squares) vs. mean interparticle spacing (r̄). Red solid line is a theoretical fitting.

In addition to the spectral peak shifting and width broadening, the scattering intensity also depends on the interparticle spacing, as revealed in Eq. (6). Figure 4 shows the measured average scattering intensity at ΩT per nanorod, I*(ΩT), as a function of r̄, as well as the fitting of Eq. (6) to the data. Again, both ωp and τ obtained from the fitting, as listed in the fifth column of Table 1, are in agreement with the ones derived from the spectral peak shifting and line-width broadening data. As shown in Eq. (6), I*(ΩT) is proportional to (ΩT/Γ)2 and therefore to Q 2. Because the transverse mode exhibits red shifting and its line width shows broadening as r̄ decreases, I*(ΩT) is expected to decreases monotonically with the decrease in r̄, which is consistent with the experimental data shown in Fig. 4. It indicates that the electric field energy stored in the system dissipates faster for smaller interparticle spacings. This r̄-dependent energy dissipation behavior can be attributed to the fact that the coupling between dipoles facilitates the energy flow between them, increasing energy loss channels due to electron-phonon scattering, inherent in τ. In addition, as pointed out previously, the Q factor reflects the effective electric field enhancement factor of the system at the resonance frequency. For the applications of using these nanorod arrays in surface-enhanced Raman scattering (SERS) [8

8. H. H. Wang, et al., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. 18, 491–495 (2006). [CrossRef]

], the dependence of Q on r̄ provides direct indication how the electromagnetic enhancement factor varies with the interparticle spacing. Since the resonance frequency is varied with the interparticle spacing simultaneously, the actual enhancement factor also depends on the laser wavelength used in Raman scattering. The comprehensive model developed here can thus provide a design tool for SERS-substrates based on plasmonic coupling in nanoparticle arrays.

Table 1. Comparison of the fitted plasma frequency, ωp, and relaxation time, τ, of the Drude model from the optical constant, ε, of Ag (Ref. 22) and those from the variation of the spectral peak shifting, ΩT, the spectral width broadening, ΓL, and the peak intensity, IT), based on Eqs. (4)-(6). The errors represent the fitting error. The analysis of the fitting results is given in Appendix.

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The experimental results of the spectral peak shifting, the line width broadening and the peak intensity variation have been shown to fit well with the developed model based on dipole coupling approximation, Eqs. (4)-(6). A question then emerges: Why does the simple dipole-coupling model work so well in our case given the concern that other multipolar contributions may also be significant? The answer to the question may first lie on the fact that the size of the nanorods is small enough and the shape is smooth enough that their polarizability can be simply represented under quasi-static approximation. Two corrections have been proposed to remedy the simplified approximation. Draine has first pointed out that the dipole polarizability needs to include a radiative-reaction correction (a k3 term) to account for energy conservation [26

26. B. T. Draine, “The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

]. Later on, another k 2 correction term was considered by many groups [27

27. M. J. Collinge and B. T. Draine, “Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry,” J. Opt. Soc. Am. A 21, 2023–2028 (2004). [CrossRef]

, 28

28. S. Zou and G. C. Schatz, Response to comment on “Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,” J. Chem. Phys. 102, 122 (2005).

]. Considering the dimension of the nanorods and the wavelengths of interest in our case, both correction terms only make ~3% difference from the uncorrected polarizability. Both of the two terms therefore only play an insignificant role in the analysis of the far-field scattering spectra. Furthermore, the insignifant contribution of the radiative-reaction (k3 term) indicates that the decrease of the Q factor with the decrease in r is caused mainly by the ohmic loss due to the electron-phonon scattering instead of the radiative loss. Another issue of concern is, for a gap as small as 5 nm (r̄=30 nm), whether the dipole approximation is still valid and therefore other multipolar effects should be included in the analysis. In a recent study of two coupled silver nanospheres, Khlebtsov and coworkers compared the difference in the spectral peak shifting between the calculated results with an electrodynamic dipole approximation as well as a more complete electrodynamic multipole solution [29]. Their results indicate that large deviation in the spectral peak shifting only occurs for r̄/R< 2.5, which is consistent with our observation. The simple analytical formulae, Eqs. (4)-(6), are therefore valid in the variation range of the interparticle spacing used in this study.

5. Conclusions

Appendix

A. SEM and TEM images

Appendix Fig. 1. Top view scanning electron microscopy image of Ag/AAO substrate with a mean diameter of 25 nm and a mean interparticle spacing of 30 nm. Inset shows the corresponding cross-section transmission electron microscopy image.

B. Derivation of the formula

The Ag nanoparticle arrays in our study can be considered as two-dimensional hexagonal arrays made of Ag prolate spheroids in AAO matrix. The transverse-mode polarizability of a single prolate spheroid along its short axis in an electrostatic field is given by [20

20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York,1983), pp. 130.

]

αT(ω)=R2hε(ω)εm3εm+3L[ε(ω)εm]
(1)

where R and 2h are the radius and the length of the spheroid. L=121e22e2(1+12eln1+e1e), where e 2=1-(R/h)2. εm is the dielectric function of the surrounding medium and

ε(ω)=1ωp2ω(ω+iτ)
(2)

is the dielectric function of the metal particle based on Drude model. ωp and τ are the plasma frequency and relaxation time, respectively. For silver in the visible wavelength range (400 to 800 nm), ωp~1016 rad/sec and τ ~10-14 sec [30

30. C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, “Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles,” J. Phys. Chem. B, 105, 2264–2280 (2001).

]. Since ω≫1/τ, Eq. (1) can be approximated as

αT(ω)(S31+Cεm)[(εm1)ω2+ωp2]ωp21+Cεmω2iωτ
(3)

where S 3=R 2 h/3L and C=(1-L)/L. It indicates that each particle behaves as a harmonic oscillator (Lorentz model) with a resonance frequency ωp(1+Cεm) and a damping constant 1/τ. In the case of arrays of particles, the local field Eloc(ri) on each particle is the sum of the incident field Einc(ri) and the fields induced by the rest of particles and can be written as [31

31. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1999), pp. 410.

]

Eloc(ri)=Einc(ri)+Σjieikrijrij3[k2rij×(rij×Pj)+1ikrijrij2×{rij2Pj3rij(rij·Pj)}]
(4)

where rij is the difference between the position vectors of i -th and j -th particles. Assuming that the local field at each particle is the same, i.e. Eloc(ri)=Eloc(rj), and considering Pj=αTEloc(rj), the effective polarizability of the arrays can be derived accordingly:

αeff=αT1αTU
(5)

where

U=Σji[k2sin2θijrijik(3cos2θij1)rij2+(3cos2θij1)rij3]eikrij
(6)

and θij is the angle between rij and Einc(ri). Notice that U depends on the interparticle spacing and the packing configuration of the array. In this study, the Ag nanorod array has a hexagonal packing pattern. With the substitution of Eq. (3), Eq. (5) becomes

αeff=S3[(εm1)ω2+ωp2][(1S3A)ωp2{(1+Cεm)+(εm1)S3A}ω2]i[(εm1)S3Bω2+(1+Cεm)(ωτ)+S3Bωp2]
(7)

where A and B are defined as U=A+iB. The scattering intensity is then the sum of the scattered radiation from each particle with this effective polarizability. This equation exhibits a quasi-Lorentzian form. The resonance peak in a scattering spectrum emerges as the real part of the denominator vanishes and, therefore, the full-width at half-maximum can be solved accordingly. It also reveals that A is responsible for plasmon resonance peak shift, while B plays a role in the linewidth broadening and is also a function of the interparticle spacing of the array, r. Considering long-wavelength approximation, i.e. exp(ikr)≈1, Eq. (6) becomes

UF·r3+iB
(8)

where F is the lattice sum of the hexagonal particle array. On the other hand, under the condition of neglecting the far-field term,

B=ji(3cos2θij1)[sin(krij)rij3kcos(krij)rij2] . The introduced error in the real part of U for the wavelength of 550 nm is smaller than 0.5% for the interparticle spacing range considered in this study. The leading term of B with a dependence on the interparticle spacing is proportional to k 5 r 2 ij which is zero under the long-wavelength approximation. This approximation is therefore not applied the subsequent derivation of B. From Eq. (7), we can derive the resonance frequency, ΩT, the corresponding linewidth, Γ, and the scattering intensity at ΩT, IT). According to Eq. (7), ΩT, corresponding to the maximum value of αeff, can be extracted by solving [(1-S 3 A) ω 2 p-{(1+m)+(εm-1)S 3 A2 T]=0 and is given by

ΩT2=ωp2[1F(Sr)3](1+Cεm)+(εm1)F(Sr)3.
(9)

Γ can be determined by evaluating the frequency at half of the maximum value of |αeff|2 and is given by

Γ=1τ+S3Bωp2(1+Cεm)ΩT+εm11+CεmS3BΩT.
(10)

Finally, IT) can be expressed as

I(ΩT)N(r)ΩT4αeff(ΩT)2=N(r){ΩTS3[(εm1)ΩT2+ωp2]Γ(1+Cεm)}2,
(11)

where N(r) is the surface dipole density. To illustrate the physical contents of the spectral peak and width, we can consider the arrays of spherical particles in air. In this case, εm=1, S=R and C=2. Equations (9)(11) then become

ΩT=(ωp3)1F(Rr)3,
(12)
Γ=1τ+R3Bωp23ΩT,
(13)

and

I(ΩT)N(r)R6ωp4(ΩTΓ)2.
(14)

In the long-wavelength approximation, B becomes zero and Eq. (12) and the resultant line width above are consistent with the derived result by Persson and Liebsch [17

17. B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).

].

C. Voigt linewidth

The dependence of the spectral peak shifting on the interparticle spacing provides a means to incorporate the influence of the variation in the interparticle spacing, presented in Fig. 1, in the analysis of the spectral width broadening data. According to Eq. (9), the dependence of the variation in ΩT, ΔΩT, on the variation in r, Δr, can be derived as

ΔΩT=32F2S3εm(C+1)ΩTr6[1+Cεm+(εm1)F(Sr)3]2Δr
(15)

and were used to calculate the inhomogeneous contribution based on the result in Fig. 1 and the extracted parameters from the spectral peak-shift analysis in the main text. The extracted Voigt width, ΓV, and r̄, is shown in the figure below.

Appendix Fig. 2. Voigt line width (ΓV) vs. mean interparticle spacing (r̄).

D. Fitting analysis

The fitted results of the spectral peak shifting vs. interparticle spacing with Eq. (9) are shown in Table 1 in the main text. The extracted plasma frequency, ωp, is just 30% away from the one obtained by fitting the optical constant [24

24. E. D. Palik, Handbook of Optical Constants of Solid (Academic Press, London,1985), pp. 353.

] of Ag to the Drude model within the wavelength range of interest. In the fitting, εm was chosen to be the dielectric constant of alumina [32

32. D. W. Thompson, “Optical characterization of porous alumina from vacuum ultraviolet to midinfrared,” J. Appl. Phys. 97, 113511 (2005). [CrossRef]

], which is 3 in our case. Since C shows 5% variation for h varied from 25 to 100 nm, C was fixed at 1.2 based on R=12.5 nm and h=50 nm. The fitted results of the spectral broadening with Eq. (10) vs. interparticle spacing are listed in Table 1 in the main text. The extracted relaxation time, τ, is about 15% higher and ωp is only 10% less than the one obtained by fitting the optical constants of Ag to the Drude model. Finally, the fitted results of the spectral peak intensity with Eq. (11) vs. interparticle spacing yield ωp and τ which are listed in Table 1 in the main text. Both ωp and τ match closely with the values obtained from the fitting of the optical constants of Ag with Drude model. In summary, these fitted values comparably agree among themselves and with the values extracted from the dielectric constants of Ag, supporting the fitting process and Eqs. (9)-(11) used in the fitting.

Acknowledgment

Authors thank the financial supports (NSC-96-2120-M-001-002 and 95-3114-P-001-007-MY3) from the National Science Council and Academia Sinica in Taiwan.

References and links

1.

M. Faraday, “On the color of colloidal gold,” Phil. Trans. R. Soc. London 147, 145–181 (1857). [CrossRef]

2.

G. Mie, “Beitrage zur optik truber medien speziel kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908). [CrossRef]

3.

M. Geissler and Y. Xia, “Patterning: Principles and some new developments,” Adv. Mater. 16, 1249–1269 (2004). [CrossRef]

4.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

5.

H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer-Verlag, Berlin,1980), pp. 116.

6.

M. Moskovits, “Surface enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985). [CrossRef]

7.

J. C. Hulteen and R. P. Van Duyne, “Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces,” J. Vac. Sci. Technol. A 13, 1553–1558 (1995).

8.

H. H. Wang, et al., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. 18, 491–495 (2006). [CrossRef]

9.

S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997). [CrossRef] [PubMed]

10.

K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. 3, 1087–1090 (2003). [CrossRef]

11.

L. Gunnarsson, T. Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, “Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions,” J. Phys. Chem. B 109, 1079–1087 (2005).

12.

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]

13.

C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” J. Phys. Chem. B 107, 7337–7342 (2003).

14.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003). [CrossRef]

15.

J. P. Kotmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express 8, 655–663 (2001). [CrossRef]

16.

X. -D. Xiang, et al., “A Combinatorial Approach to Materials Discovery,” Science 268, 1738–1740 (1995). [CrossRef] [PubMed]

17.

B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).

18.

V. A. Markel, “Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure,” J. Mod. Opt. 40, 2281–2291 (1993). [CrossRef]

19.

see Appendix.

20.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York,1983), pp. 130.

21.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 371.

22.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1999), pp. 407.

23.

Y. Liu, J. Lin, G. Huang, Y. Guo, and C. Duan, “Simple empirical analytical approximation to the Voigt profile,” J. Opt. Soc. Am. B 18, 666–672 (2001).

24.

E. D. Palik, Handbook of Optical Constants of Solid (Academic Press, London,1985), pp. 353.

25.

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt linewidth: A brief review” J. Quant. Spectrosc. Radiat. Transfer 17, 233–236 (1977). [CrossRef]

26.

B. T. Draine, “The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

27.

M. J. Collinge and B. T. Draine, “Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry,” J. Opt. Soc. Am. A 21, 2023–2028 (2004). [CrossRef]

28.

S. Zou and G. C. Schatz, Response to comment on “Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,” J. Chem. Phys. 102, 122 (2005).

29.

B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches,” Nanotech. 17, 1437–1445 (2006). [CrossRef]

30.

C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, “Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles,” J. Phys. Chem. B, 105, 2264–2280 (2001).

31.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,1999), pp. 410.

32.

D. W. Thompson, “Optical characterization of porous alumina from vacuum ultraviolet to midinfrared,” J. Appl. Phys. 97, 113511 (2005). [CrossRef]

OCIS Codes
(290.5850) Scattering : Scattering, particles
(160.4236) Materials : Nanomaterials

ToC Category:
Scattering

History
Original Manuscript: March 31, 2008
Revised Manuscript: April 28, 2008
Manuscript Accepted: May 15, 2008
Published: September 15, 2008

Citation
Sajal Biring, Huai-Hsien Wang, Juen-Kai Wang, and Yuh-Lin Wang, "Light scattering from 2D arrays of monodispersed Ag-nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling," Opt. Express 16, 15312-15324 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15312


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References

  1. M. Faraday, "On the color of colloidal gold," Phil. Trans. R. Soc. London 147, 145-181 (1857). [CrossRef]
  2. G. Mie, "Beitrage zur optik truber medien speziel kolloidaler metallosungen," Ann. Phys. 25, 377-445 (1908). [CrossRef]
  3. M. Geissler and Y. Xia, "Patterning: Principles and some new developments," Adv. Mater. 16, 1249-1269 (2004). [CrossRef]
  4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424,824-830 (2003). [CrossRef] [PubMed]
  5. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer-Verlag, Berlin, 1980), pp. 116.
  6. M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985). [CrossRef]
  7. J. C. Hulteen and R. P. Van Duyne, "Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces," J. Vac. Sci. Technol. A 13, 1553-1558 (1995).
  8. H. H. Wang,  et al., "Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps," Adv. Mater. 18, 491-495 (2006). [CrossRef]
  9. S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102-1106 (1997). [CrossRef] [PubMed]
  10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003). [CrossRef]
  11. L.  Gunnarsson, T.  Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).
  12. P. K. Jain, W. Huang, and M. A. El-Sayed, "On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation," Nano Lett. 7, 2080-2088 (2007). [CrossRef]
  13. C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
  14. W.  Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003). [CrossRef]
  15. J. P. Kotmann and O. J. F. Martin, "Plasmon resonant coupling in metallic nanowires," Opt. Express 8, 655-663 (2001). [CrossRef]
  16. X. -D. Xiang,  et al., "A Combinatorial Approach to Materials Discovery," Science 268, 1738-1740 (1995). [CrossRef] [PubMed]
  17. B. N. J. Persson and A. Liebsch, "Optical properties of two-dimensional systems of randomly distributed particles," Phys. Rev. B 28, 4247-4254 (1983).
  18. V. A. Markel, "Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure," J. Mod. Opt. 40, 2281-2291 (1993). [CrossRef]
  19. see Appendix.
  20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 130.
  21. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 371.
  22. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 407.
  23. Y. Liu, J. Lin, G. Huang, Y. Guo, and C. Duan, "Simple empirical analytical approximation to the Voigt profile," J. Opt. Soc. Am. B 18, 666-672 (2001).
  24. E. D. Palik, Handbook of Optical Constants of Solid (Academic Press, London, 1985), pp. 353.
  25. J. J. Olivero and R. L. Longbothum, "Empirical fits to the Voigt linewidth: A brief review" J. Quant. Spectrosc. Radiat. Transfer 17, 233-236 (1977). [CrossRef]
  26. B. T. Draine, "The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
  27. M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004). [CrossRef]
  28. S. Zou and G. C. Schatz, Response to comment on "Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,"J. Chem. Phys. 102, 122 (2005).
  29. B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches," Nanotech. 17, 1437-1445 (2006). [CrossRef]
  30. C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, "Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles," J. Phys. Chem. B,  105, 2264-2280 (2001).
  31. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 410.
  32. D. W. Thompson, "Optical characterization of porous alumina from vacuum ultraviolet to midinfrared," J. Appl. Phys. 97, 113511 (2005). [CrossRef]

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