Article pubs.acs.org/JPCC
Structural Effects in the Electromagnetic Enhancement Mechanism of Surface-Enhanced Raman Scattering: Dipole Reradiation and Rectangular Symmetry Effects for Nanoparticle Arrays Logan K. Ausman, Shuzhou Li, and George C. Schatz* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *
ABSTRACT: Surface-enhanced Raman scattering (SERS) enhancement factors for Ag and Au sphere array structures are determined by rigorously including dipole reradiation in a T-matrix formalism. Comparisons are made with the more commonly used local field enhancement due to plane-wave excitation, |E(r0;ω)|2|E(r0;ωs)|2 which for zero Stokes shift is |E(r0;ω)|4 to determine the errors associated with this approximation. Substantial errors (factors of 10−100) are found for the peak enhancements at a scattering angle well away from the incident direction, but for backscattering, the errors are negligible. We also present |E|4 enhancement factors using a periodic boundary discrete dipole approximation method for several metal strip array structures, and we show that a certain combination of rectangular array structure and strip properties leads to electromagnetic enhancement factors for mixed photonic-plasmonic resonances that are considerably higher than can be produced with either square arrays or 1-D arrays based on the same particles and spacings.
I. INTRODUCTION
Past interest in nanoparticle array structures has included studies of 1-D linear chains and 2-D arrays that are both irregular and regular.25−44 In the papers by Zou and Schatz, linear chains of spheres and sphere dimers were studied using both T-matrix theory and the coupled dipole approximation. They found that long-range photonic resonances in the 1d array led to narrow features in the extinction spectrum that are associated with a large enhancement in the electric field near the particles. In one paper,45 they observed that these photonic resonances (or photonic lattice modes) added an additional factor of 17 to the enhancement over the isolated particle case. Zhao et al. conducted a similar study of EFs for a 1d array of silver nanoshell dimers.31 They observed an additional order of magnitude enhancement of the optimal dimer array with respect to an isolated dimer. This work was later extended to 2d arrays of nanoshell dimers by Song et al.33 where the results were similar to the 1d array findings. Optimization of the EFs for mixed photonic-plasmonic resonances through the use of rectangular arrays and anistropic nanoparticles has not been considered. Because the majority of classical electrodynamic methods involves the scattering of an incident plane polarized wave it is not surprising that the existing models of SERS due to substrates made of arrays of nanoparticles have been based on the planewave (PW) approximation.27,28,45,46 This approximation assumes that the SERS EF is the product of the electric field intensity at the incident frequency, |Eloc(r0;ω)|2, and the electric field intensity at the Stokes shifted frequency, |Eloc(r0;ωs)|2, both
Since the discovery of surface-enhanced Raman scattering (SERS) in the 1970s1−3 there has been much interest in the design and fabrication of SERS substrates that optimize the enhancement factor (EF). This has led to applications in biomolecule4−8 and chemical sensing9−11 with specific examples in the analysis of anthrax,5,7 half-mustard agent,9 pigments in art restoration,10,11 and glucose.4,6,7 Further improvements in substrate fabrication have been stimulated by the discovery of single-molecule SERS.12−14 The prospect of employing SERS in sensing applications has resulted in interest in nanoparticle array structures that support SERS, as such structures are readily fabricated by a variety of techniques, including soft-lithography methods,15 that lend themselves to mass production. However the design of viable array structures to be used as SERS sensors requires a thorough understanding of the electromagnetic interactions between nanoparticles. It is widely accepted that a major component of the SERS enhancement is the result of locally enhanced electric fields produced by the substrate at the location of the molecule. The most common substrates are metal nanoparticles, usually Au or Ag, and the local electric field enhancements arise because of localized surface plasmon resonance (LSPR) excitation6,16−19 in these particles when they interact with light. Because the LSPR is a collective oscillation of conduction electrons, it behaves largely according to the laws of classical electrodynamics. Therefore, it makes sense that most models of SERS are purely classical, although there are examples in the literature that incorporate quantum mechanical methods.20−24 © 2012 American Chemical Society
Received: December 20, 2011 Revised: July 12, 2012 Published: July 12, 2012 17318
dx.doi.org/10.1021/jp2122938 | J. Phys. Chem. C 2012, 116, 17318−17327
The Journal of Physical Chemistry C
Article
evaluated at the molecular location. This means that |Eloc(r0;ω)|2 and |Eloc(r0;ωs)|2 are obtained from calculations that involve PW scattering from the substrate. Often an additional assumption is made that the plasmon resonance occurs over a broad range of frequencies and that for small Stokes shift the enhancement can be taken to be |Eloc(r0;ω)|4; in this article we refer to this case as the PW approximation. An infinite PW is a reasonable model for most sources of illumination in real experiments; however, the reradiation field by the dipole induced in the molecule at the Stokes shifted frequency can be considerably different from that of a PW. Because the molecule is much smaller than the metal particles it interacts with, it is appropriate to model this field as that arising from a point dipole. There have been a few studies that incorporate dipole reradiation (DR)47−51 rigorously into the evaluation of electromagnetic EFs, but the great majority including all studies of nanoparticle arrays have used the PW approximation. However the comparisons between DR and PW results show significant errors in the PW approximation for certain scattering geometries, providing motivation for doing further studies of DR effects. For example, for a spherical dimer system,52 it was observed that the agreement between PW and DR results is highly dependent on both detector and molecule location. These differences were found to depend on interferences between the directly emitted field and the field that scatters from the nanoparticles and on effects that arise from excitation of higher order angular momentum components (multipole resonances) in the plasmon excitation of the nanoparticles. It can be expected that similar but more complicated processes will play a role in SERS on array structures, but this has not been considered so far. Here we model SERS for a molecule in the presence of a substrate comprised of an array of Ag or Au nanospheres by employing a T-matrix method that rigorously incorporates DR, and we compare this with the corresponding results of PW calculations. From these results, we are able to establish when the PW approximation is adequate and when not. On the basis of these results, we then use the PW approximation within the context of the discrete dipole approximation (DDA) approach with periodic boundary conditions, and we show how the parameters of the DDA calculation may be chosen such that DDA and T-matrix results are in good agreement. Then, with the periodic DDA method, we study mixed photonic/plasmonic resonances for 1-D and 2-D metal strip array structures, and we find that for rectangular arrays with a 2:1 ratio of array dimensions the SERS EFs are larger than those for square arrays and considerably larger (factor of >10) than can be obtained from 1-D arrays or from isolated strips. Chemical contributions to the SERS EF have been ignored in this work, and the arrays are assumed to be in a homogeneous dielectric, such as can be achieved by index matching if the array is on a surface. The T-matrix and DDA theories are outlined in Section II, whereas Section III presents the results for arrays of spheres and for rectangular metal strip arrays. In Section IV we present some concluding remarks.
theory framework, which employs a basis of vector spherical harmonic functions to represent an arbitrary field. That is ∞
E=
l
[plm N(lmσ )(k r) + qlm M(lmσ )(k r)]
∑∑ l = 1 m =−l
(1)
with coefficients plm and qlm, and vector spherical harmonic basis functions N and M that are solutions of the vector Helmholtz equation and are identically 0 for l = 0. In spherical polar coordinates these basis functions are N(lmσ )(k r) =
1 ∇ × M(lmσ )(k r) k
(2)
(σ ) M(lmσ )(k r) = ∇ × {rulm (k r)}
(3)
and the scalar function ulm(kr) satisfies the scalar Helmholtz equation. In spherical polar coordinates this is (σ ) ulm (k r) = zl(σ )(kr )Plm(cos θ )eimϕ
zl(σ )(kr )
⎧ j (kr ), for σ = 1 ⎪l =⎨ ⎪ h (1)(kr ), for σ = 3 ⎩ l
(4)
The function zl is a spherical Bessel function that is chosen to satisfy the appropriate boundary conditions of the scattering problem; that is, jl is a spherical Bessel function and hl(1) is a spherical Hankel function. The Pml are associated Legendre functions and together with the exponential these form the scalar spherical harmonics. To incorporate a dipole electric field source in a vector spherical harmonic basis, we begin by noting that the electric field of an oscillating induced dipole, p, can be obtained by using the relation E(r; ω) =
4πiω c2
∫ G0(r, r′; ω)·J(r′; ω) d3r′
=
4πk 2 εμ
∫ G0(r, r′; ω)·pδ(r′ − r0) d3r′
=
4πk 2 G0(r, r0; ω) ·p εμ
(5)
where in the above G0 is the free-space tensor Green’s function that can be obtained in a vector spherical harmonic basis via the Ohm-Rayleigh method56 and is G0(r>, r)N l ± m(k r)M l ± m(k r| > |r
