Optical properties of ultrafine gold particles

We have prepared spherical and individually isolated gold particles with median diameters in the interval 3 to 4 nm and log-normal size distribution by evaporation in a few Torr of air. The volume fraction of metal was typically 0.3%. Highly reproducible optical transmittance in the range 0.3 to 4 \ensuremath{\mu}m was recorded provided the deposits (onto glass substrates) were sufficiently thin to prevent coagulation into large aggregates. A large transmittance minimum was always found at about 0.6 \ensuremath{\mu}m, whereas the samples were practically transparent at wavelengths above 1.5 \ensuremath{\mu}m. To interpret the measurements we have performed extensive computer calculations based on the effective-medium theories by Maxwell-Garnett (as extended by Polder and van Santen), Bruggeman, and by Hunderi. These theories are indistinguishable in the limit of small filling factors. As input data we have used bulk results, for which the Drude part was modified to account for a size-limited electron mean free path assuming diffuse boundary scattering. The calculated wavelength-dependent transmittances displayed a minimum whose distinctness was strongly dependent on the median particle diameter, whereas the distribution width or shape played a subordinate role. The computed results could be brought into fair agreement with the experiments provided we invoked an apparent plasma frequency for the particles which was lower than the bulk value and an effective sample thickness of the order of twice the measured one. To explain the meaning of this seemingly ad hoc procedure we have calculated (using unshifted plasma frequencies and sample thicknesses) the effects of a dielectric coating on the particles and of deviations from nonspherical shapes. Neither can be reconciled with all the experimental evidence. We finally considered the role of dipole-dipole coupling which is significant despite the small filling factors because the gas evaporated particles must be touching. This interaction is accounted for approximately by a set of effective depolarization factors pertaining to each of a number of well-defined geometrical configurations of spheres. Assuming the deposits to consist of a mixture of close-packed clusters, infinite linear chains, and independent single particles (i.e., by use of two adjustable parameters solely, whose magnitudes were supported by electron micrographs), we were able to reproduce the experimental transmittance data to within a few percent over the entire wavelength interval. The remaining deviations were presumably caused by our simplified treatment of the dipole-dipole interaction, and hence there appears to be no effects due to size quantizations.