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

5.2. 2-DIMENSIONAL ARRAYS OF GOLD NANOPARTICLES 81contribution of the bulk LiF crystal and the deposited LiF; since we found variationsof the refractive indices within the 1% of the absolute values, here we assume anhomogeneous and isotropic material, with optical constants reported in fig. 5.2.The “film”, instead, represents the surface layer accomodating the nanoparticles.In the previous section we treated the ripples as cylinders, now, however, as theripple birefringence is much lower than the optical anisotropy of the nanoparticles,we apply a much simpler approximation, describing the ripples as an homogeneouseffective host, with dielectric constant that are a linear combination of the opticalconstants of LiF and air. We also employ effective medium approximation for theAu nanostructures, calculating an anisotropic effective dielectric tensor ε eff for the“film”, which includes the contributions of the NPs shape and mutual EM coupling(see eq. (5.14)).zyxd zd yd xa zΛa.d effε ma yε hε sa xd effεeffb. ε sc.Figure5.5: SchematicrepresentationoftheapproximationsappliedtothegoldNPs. Panela: gold NPs supported on a perfect and regular LiF ripple structure, and arranged on arectangular mesh. Panel b: the NPs are assumed ellipsoidal shaped, with axes aligned tothe plane of the sample, and immersed in an homogeneous and isotropic host. Panel c:the host and the NPs are replaced by an homogeneous effective medium with anisotropicdielectric constant ε eff .Under these assumptions, we model the Au/LiF nanostructures as an ensemble ofN ≫ 1 identical and aligned polarizable inclusions, immersed in an homogeneous andisotropic host and placed on top of an homogeneous and isotropic substrate, as sketchedin fig. 5.5(b). We set the coordinate system with the axis z normal the substrate andthe axes x and y on the surface, the axis y being parallel the ripples direction. All theinclusions have the same ellipsoidal shape, with principal semiaxes a x , a y and a z along thecartesian axes, and are arranged on a rectangular grid with first-neighbour spacings d xand d y ; the center of the inclusions is placed at distance d z > a z from the substrate/filminterface. The substrate, the host medium and the inclusions have, respectively, dielectricfunctions ε s (ω), ε h (ω) (both purely real) and ε m (ω) (complex), reported in the graphsin fig. 5.6. ε s is converted from the refractive index of bulk LiF (fig. 5.2), while ε his the average between ε s and vacuum (ε = 1); ε m is instead obtained by extractingthe dielectric constant of gold from ellipsometric measurements performed on clean bulksamples (Au(111)/Mica from PHASIS, inset of fig. 4.4), and then applying the finite sizecorrections according to eq. (1.57).In the quasi-static approximation (§1.3.1) the dipolar polarizability tensor α of theindividual inclusions is diagonal and its principal components are given by eqs. (1.60)and (1.42). In order to account for the morphological disorder of the nanoparticles, thesize/shape dispersion unavoidably encountered in our samples are effectively convertedinto a corresponding distribution of depolarization factors L γ , and accordingly replaceeq. (1.60) with [208]: