Morphology and plasmonic properties of self-organized arrays of ...

22 CHAPTER 1. THEORYε MG −ε aε MG +2ε a= f bε b −ε aε b +2ε a(1.38)This and the equivalent equation for f a ≪ f b are the Maxwell-Garnett effective mediumequations [130, 131]. They are usually applied when dealing with systems composed ofwell isolated particles or grains dispersed in continuous media (fig. 1.8(a)).In cases where f a and f b are comparable, it may not be clear the distinction betweenhost and inclusions. Then, we can make the self-consistent choice ε = ε h , and (1.37)reduces toε a −ε Br0 = f aε a +2ε Br +f ε b −ε Brbε b +2ε Br (1.39)This is the Bruggeman expression [132], commonly known as effective medium approximation(EMA)andsuitedtodescribeuniformmixturesoftwodifferentmaterials(fig.1.8(b)).The above effective medium expressions are few of the simplest approximations fordescribing the optical constants of heterogeneous media. One of the main assumptionsis that all the phases feel the same equivalent mean field, implying that the domains areuniformly distributed in the volume of the medium. This is not the case, for example,when the grains are coherently arranged on a lattice [133, 134], so that the local fielddistributionhasthesamesymmetryofthelattice, orarestronglycoupledbyEMradiation[135, 136], so that the local field is highly localized (see §1.3). Another situation wheresuch EMAs fail is the proximity of percolation, when long-range conductive paths areestablished between the grains of the single phases [137–139]. The application of effectivemedium theories must be therefore carefully evaluated depending on the specific caseunder scrutiny, taking into account that they are always a simplification of heterogeneoussystems into equivalent single-phase media.1.3 Optical properties of metallic nanostructuresThe optical properties of metals, from low energies up to the near-UV, are mostly dominatedby the contribution of the free electrons, which are weakly bound to the metallicatoms and can freely move inside the crystal. These electrons, for example, are responsiblefor the high electric conductivity and the high optical reflectivity and absorption from DCup to visible and near UV frequencies.On the other hand, metallic nanostructures with sub-micrometer dimensions exhibitvery different optical responses with respect to their bulk counterparts. An externalEM field can penetrate inside the volume of the particles, shifting the free electrons gaswith respect to the ions lattice; consequently, charges of opposite sign accumulate on theopposite surfaces of the particles, polarizing the metal and establishing restoring localfields (E R in fig. 1.9). Therefore, in formal analogy with the Lorentz model, the particlescan be viewed as oscillators, whose behaviour is determined by the free electrons effectivemass, charge and density, but most importantly by the geometry of the particles [62–64, 66, 140]. Under resonance conditions, the free electrons gas is coherently draggedby the external excitation, so the electric dipoles induced inside each particles becomeextremely large. Correspondingly, the local fields in proximity of the particles are orderof magnitudes enhanced with respect to the incident fields, the scattering cross sectionis enormously amplified, and very strong absorption peaks are observed. Such collectiveexcitations are commonly known as localized surface plasmons (LSPs) [54, 61–65]; for