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

28 CHAPTER 1. THEORYwhere Γ 0 = v F /λ mfp is determined by the electrons mean free path in the metal bulk (v Fis the Fermi velocity). Surface damping can be empirically modeled [151] as an additionalsize-dependent contribution Γ surf (a) to Γ 0 , which writesΓ 0 (a) = Γ 0 +Γ surf (a) = v Fλ mfp+A v Fa . (1.54)A is an empirical factor, of the order of 1, which incorporates the details of the scatteringprocesses [61]; for a sphere it is usually chosen between 3/4 and 1 [62, 63, 152, 153].Introducing Γ 0 in ε intra , we obtainε intra (ω,a) = 1−ω 2 Pω 2 −iΓ 0 (a)ω(1.55)Now, we can modify the dielectric constant by subtracting from (1.52) the bulk intrabandcontribution (1.53) and adding the corrected term (1.55): we find the size-dependentε m (ω,a) given byε m (ω,a) = ε exp (ω)−ε intra (ω)+ε intra (ω,a)= ε exp (ω)+∆ε(ω,a)(1.56)with [154]∆ε(ω,a) = ω2 Pω( )1 1−ω −iΓ 0 ω −iΓ 0 (a)(1.57)In fig. 1.13, the dielectric constants of gold nanoparticles with radius a = 5 nm and50 nm are compared. We can see that reducing the particle size the imaginary part ofε m increases at the larger wavelengths (lower energies), while the real part is only slightlyraised. Then, according to the Fröhlich condition (1.45), the resonances do not experiencesignificant shifts, and the main effects of surface damping is a broadening of the plasmonslinewidth, in accordance with the experimental results [61].Retardation effectsIn the previous sections we have seen that in the quasi-static approximation the incidentEM radiation is considered uniform within the particles volume. This assumption can beadequate for particles with sizes up to 100 nm and EM frequencies in the visible range,however it fails to predict the dependence of the optical response on the NP dimensions.Moreover, as the variations of the incident EM fields cannot be neglected anymore, multipolareigenmodes are also excited. Nevertheless, if the NPs sizes are lower than ≈ 10%of the typical wavelenght of the incident radiation [155] (≈ 40 nm in the visible range),multipolar contributions can still be neglected and retardation effects can be explicitlycalculated for the dipolar modes [156]; this is sometimes called modified long-wavelengthapproximation (MLWA). For gold ellipsoidal nanoparticles, it has been shown that it consistentlyreproduces exact numerical solutions for particles with equivalent volumes upto a ≈ 40 nm radius sphere and aspect ratios below 10 [66, 157, 158], and it is still inqualitative agreement with results at ≈ 200 nm dimensions [159]. As the particles underscrutiny in this thesis have typical sizes of several tens of nanometers, we will also employMLWA for calculating their polarizabilities.Let’s consider a spherical particle with radius a excited by an EM field. In QSA wefound that the induced dipole p is proportional to the incident electric field (1.47). The