Nanosphere Lithography:  Surface Plasmon Resonance Spectrum of a Periodic Array of Silver Nanoparticles by Ultraviolet−Visible Extinction Spectroscopy and Electrodynamic Modeling

In this paper we measure the optical extinction spectrum of a periodic array of silver nanoparticles fabricated by nanosphere lithography (NSL) and present detailed comparisons of the results with predictions of electrodynamic theory. The silver nanoparticles are small (∼100 nm) compared to the wavelength of light but too large to have their optical properties described adequately with a simple electrostatic model. We make use of the discrete dipole approximation (DDA), which is a coupled finite element method. With the DDA one can calculate the extinction of light as a function of wavelength for particles of arbitrary size and shape. We show that NSL-fabricated Ag nanoparticles can be modeled without adjustable parameters as truncated tetrahedrons, taking their size and shape parameters directly from atomic force microscopy (AFM) measurements and using literature values of the bulk dielectric constants of silver. These AFM measurements are presented as part of this paper, and the resulting theoretical line shapes and peak widths based on the AFM-derived parameters are in good agreement with measured extinction spectra. The peak width measured as the full width at half-maximum (fwhm) is approximately 100 nm, or 0.35 eV, which corresponds to an electron−hole pair lifetime of 2 fs. The combined effects of particle−particle and particle−substrate interactions red-shift the surface plasmon resonance by only about 10 nm versus a single isolated particle. By use of AFM-derived parameters that have been corrected for tip-broadening and by inclusion of an estimate for the effects of particle−particle and particle−substrate interaction, the discrepancy between the theoretical and experimental extinction peak maxima is approximately 25 nm, which is significantly smaller than the plasmon width. This residual difference between theory and experiment is due to shortcomings of the truncated tetrahedron geometry in describing the actual shape of the particles, errors in the literature values of the bulk dielectric constants, and experimental uncertainty due to slight heterogeneities in nanoparticle structure.