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

78 CHAPTER 5. MODELLING AND ANALYSIS OF THE OPT. PROP.semi-infinite homogeneous substrate (N 2 ); for simplicity we assume the refractive indicesN i as real constants (and N 0 = 1). Referring to fig. 5.3, the constructive interferenceoccurs whenever the difference between the optical paths travelled by the reflected beamsis an integer multiple m of the wavelength, i.e. [126]N 1 (BC +CD)−N 0 AD = 2dN 1 cosθ 1 = mλ m = m hcE mwhere E(λ) = hc/λ is the photon energy. The period of the interference figure (forexample as a function of E) is then given by∆E = E m+1 −E m =hc2dN 1 cosθ 1,directly related to the thickness and refractive index of the film. In this simple example,the period ∆E is independent of the energy and inversely proportional to d; in all realcases, however, this is not true due to the dispersion of the refractive indices, so a completeanalysis of the optical constants is required to estimate the thickness.In order to characterize the nanostructured substrates, we proceed as in the previouscase, building an optical model and then performing a best-fit procedure to extract theoptical constants. We extend the model employed for the bare LiF(110), including asecondCauchylayertodescribethedepositedlayer, thatrequiresopticalconstantsslightlydifferent from the substrate. Then, in order to reproduce the anisotropy of Ψ and ∆, weaddontopofthe“film”layerabirefringentsurfacelayer, havingdifferentrefractiveindicesin the directions parallel and perpendicular the ripples, which accounts for the rippledsurface morphology. An accurate treatment would require to fully consider the sawtoothlikeshape of the ripples; here, instead, we apply a simpler approximation, treating theripples as parallel cylinders aligned in the plane of the sample. This is implemented bycombining two Bruggeman effective medium layers, with depolarization factors L || = 0along the ripples and L ⊥ = 0.5 in the perpendicular directions (see fig. 1.11(c)), in abirefringent material placed on top of the film; both the EMA layers are composed inequal proportion of air and of the underlying film material. As in the previous case, weneglect any effect of surface contamination or disorder. The overall structure of the modelis sketched in fig. 5.4(c). All the components employed (EMAs, birefringent layers, etc.)are standard elements provided by the WVASE software.The best-fit Ψ and ∆ curves corresponding to the experimental spectra of a nanopatternedLiF (fig. 4.3) are reported in fig. 5.4(a,b). The best-fit thickness of the film and ofthe roughness layer, for the particular sample under scrutiny, were 249 nm and 4.2 nm,respectively, while a ripple periodicity of about 35 nm was deduced from AFM. The agreementbetween the simulations and the experiments is again not excellent, but the mainfeatures of the ellipsometric spectra are all well reproduced. The calculated curves forthe plane of incidence parallel and perpendicular the ripples direction are indeed slightlyshifted, the differences being more pronounced in ∆ than in Ψ, and in agreement with theexperiments, indicating that the approximation to cylinder is good for a first order analysis.The thickness of the effective layer was instead quite underestimated in comparisonto the AFM data; in the ideal case of a perfect ripple structure, the depth of the valleysis equal to half of the periodicity, corresponding to ≈17 nm for the current case; if wefurther include some disorder and roughness, then we would expect a thickness greaterthan ≈20 nm, which is clearly in contrast with the value of merely 4.2 nm extracted fromthe fit.