Resonant Field Enhancements from Metal Nanoparticle Arrays

Theoretical and semiempirical studies of two-dimensional (2D) metal nanoparticle arrays under periodic boundary conditions yield quantitative estimates of their electromagnetic (EM) field factors, revealing a critical relationship between particle size and interparticle spacing. A new theory based on the RLC circuit analogy has been developed to produce analytical values for EM field enhancements within the arrays. Numerical and analytical calculations suggest that the average EM enhancements for Raman scattering (Gh ) can approach 2 × 1011 for Ag nanodisks (5 × 1010 for Au) and 2 × 109 for Ag nanosphere arrays (5 × 108 for Au). Radiative losses related to retardation or damping effects are less critical to the EM field enhancements from periodic arrays compared to that from other nanostructured metal substrates. These findings suggest a straightforward approach for engineering nanostructured arrays with direct application toward surface-enhanced Raman scattering (SERS). Nanostructured metal-dielectric interfaces often exhibit enhanced optical phenomena at visible and near-infrared (NIR) frequencies via excitation of surface plasmon modes.1,2 The enticing possibilities of engineering such properties for applications in photonics and chemical sensing have led to a resurgence of activity in the design of plasmonic materials with subwavelength dimensions.3 Enhanced electromagnetic (EM) field effects can be generated either in a broad spectral range, as is the case for disordered metal-dielectric composites,2,4 or at select frequencies from periodically ordered metal nanostructures. Periodicity plays a key role in tuning the optical response of the latter, and has been documented in experimental and theoretical investigations of plasmonenhanced effects such as surface-enhanced Raman scattering (SERS),5-7 extraordinary optical transmission,8-10 and robust photonic band gaps at visible and NIR wavelengths.11-13 SERS has attracted widespread attention because of its demonstrated potential for single-molecule spectroscopy and chemical sensing with high information content.14-16 The rational design of optimized SERS substrates remains a challenging goal, despite extensive efforts to elucidate the physical basis of signal enhancement. Several theoretical studies have described highly localized EM fields at the junction of metal nanostructures,7,17-19 with local EM enhancement factors Gloc ) |Eloc(λ)/E0(λ)|4 as high as 10111012 for a two-particle system.20 However, less attention has been paid to the average EM enhancement factors (Gh ) 〈Gloc〉), which has greater relevance for the design and optimization of SERS-based chemical sensors. In this regard, Garcı́a-Vidal and Pendry have provided electrodynamics calculations on periodic nanostructured metal films with Gh values on the order of 106, a level of activity commonly observed in many experimental systems.7 Here we provide numerical calculations and a simple analytical theory for calculating EM field enhancements in two-dimensional (2D) arrays of metal nanoparticles embedded in a dielectric medium. The numerical simulations and analytical values are in good agreement and yield Gh values as high as 2 × 1011 for arrays of cylindrical Ag nanodisks and 5 × 1010 for arrays of Au nanodisks. Analytical values can also be obtained for 2D arrays of metal nanospheres, yielding Gh values on the order of 2 × 109 and 5 × 108 for Ag and Au nanoparticles, respectively. These activities, which are up to several orders of magnitude greater than those of aperiodic metal-dielectric composites or roughened metal surfaces, are independent of morphology-dependent resonances such as those responsible for microcavityenhanced SERS.21 The enhancements of the nanostructured arrays are strongly dependent on the ratio of particle diameter to interparticle spacing, which determines both the intensity of local field factors and the available cross-sectional area for sampling chemical and biomolecular analytes. Our models illustrate how the interplay between field enhancement and interparticle spacing can impact the design of arraybased SERS sensors for trace chemical analysis. Numerical calculations were first performed on 2D arrays of cylindrical oblate disks with high aspect ratio and diameter * Corresponding author. E-mail: [email protected] † Department of Electrical and Computer Engineering. ‡ Department of Chemistry. NANO LETTERS 2004 Vol. 4, No. 1 153-158 10.1021/nl0343710 CCC: $27.50 © 2004 American Chemical Society Published on Web 12/09/2003 d , λ, arranged in square or hexagonal lattices in a dielectric medium d using periodic boundary conditions (see Figure 1).22 Such arrays can be approximated as planar systems for the purpose of estimating local field factors.2 Here we apply the current conservation law, which can be expressed in terms of a local potential æ(r) where E0 is the incident electric field, σ(r) ) -iω m/4π is the local conductivity, and m ) ′m + i ′′ m is the complex dielectric function of the metal. Describing the continuity equation in these terms allows the collective plasmon response to be determined under quasistatic conditions in a scale-invariant manner.2 In addition, Eloc(r)/E0 can be calculated as a continuous function of packing density, described by a single geometric parameter γ ) d/δ, where d is the distance between the particles at the point of closest approach. The parameter γ is fundamentally important, because the resonance condition can be derived directly from Poisson’s equation to yield the simple relationship ′m ≈ -γ d. It is important to note that the quasistatic approximation is valid for nanoparticle arrays with periodicities below the diffraction limit (λ/2). Radiative loss from elastic (Rayleigh) scattering is negligible, and losses due to retardation effects (a function of the finite skin depth of nanoparticles larger than 30 nm) can be accounted for by a first-order correction (see below).2 Furthermore, for a 2D nanoparticle array where γ . 1, the spatial localizations of the EM resonances between particles are well within the quasistatic limit. Discretization of eq 1 on a square-mesh lattice under periodic boundary conditions yielded L2 equations (L ) 120), which were solved by the exact block elimination approach.23 This method provides solutions for site potentials corresponding to Eloc(r)/E0 in L4 operations, an enormous savings in computing time compared with the L6 operations required by Gaussian elimination methods.24 Gh is obtained simply as the mean value of Gloc) |Eloc(r)/E0|4 within a unit cell of the periodic lattice. We note that these calculations are equally valid for periodic arrays of nanowires at constant depth as for oblate metal nanodisks with high aspect ratio, whose electric fields and currents are confined to the plane of the system.2 The intensities of the local and average EM field enhancements depend greatly on both incident wavelength and diameter-spacing ratio (see Figures 1 and 2 for Gloc and Gh of Au nanodisk arrays at different values of γ). Au nanodisk arrays with large diameter-spacing ratios (γ g 30) can produce Gloc values as high as 1010, whereas Ag nanodisk arrays can produce Gloc values as high as 1012. These optical gains are the product of wavelength-selective resonant modes within the periodically ordered arrays (see below).22 With respect to the average EM enhancements, Gh can be described as a resonance band whose width increases with γ. Arrays with large γ can produce high Gh over a greater range of excitation wavelengths, which has important practical rami-