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

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84 CHAPTER 5. MODELLING AND ANALYSIS OF THE OPT. PROP.where S ξξ ≡ ∑ j sξξ ij e−iq·rij and r ij = r j −r i .We can now introduce the effective polarizability tensor A, which expresses the polarizabilityof the metallic inclusions accounting for their reciprocal dipolar interactions:A γγ =Then, the induced dipoles can be finally written asα γ1−ε 0 α γ Sγγ, γ = x,y,z (5.10)p = ε o ε h A⊗E ex (5.11)We can now formulate an effective medium approximation for the metallic inclusions.Given the isolated nature of the inclusions, we follow an approach similar to the Maxwell-Garnett EMA (§1.2), modified in order to account the ordered arrangement of the inclusionsand the presence of the substrate (cfr. also refs. [209–211]).We note that the induced dipoles (5.11) within the array have the same expression(1.41) for a single isolated particle, except for the polarizability tensor α replaced by A.Then, we can treat the inclusions as isolated, and write the total polarization density asP = np = ε 0 ε h nA⊗E ex , (5.12)where n = 1/d x d y d eff is the number of inclusions per unit volume, and d eff the thicknessof the host. The effective dielectric tensor ε eff for the inclusions can be evaluated fromthe definition of the electric displacement field DD = ε 0 ε eff ⊗E ex = ε 0 ε h E ex +P = ε 0 ε h E ex +ε 0 ε h nA⊗E ex (5.13)where we can see that ε eff is diagonal similar to A. Solving separately for the threeprincipal components we finally obtainε ξξeffε h= 1+nA ξξ , ξ = x,y,z (5.14)Using the dielectric tensor ε eff we can represent the inclusions and the host as an homogeneousand biaxially anisotropic film (fig. 5.5(c)), still preserving all the informationson the NPs shapes and interactions. The optical response of the whole system can thenbe easily calculated using the Fresnel coefficients (1.26) for the thin-film/substrate model.In this respect, however, the refractive indices for the s and p components of the electricfield are required, whereas the dielectric tensor ε eff is expressed in xyz coordinates. Weneed therefore to rewrite the components of ε eff in sp coordinates.In general the reflection of light from a biaxially anisotropic film is a quite complexphenomenon, especially when the principal axes of the dielectric tensor are not parallel tothe interfaces or the plane of incidence [172, 212]. In this case, however, we just need toreplicate the geometries applied in the experiments, i.e. plane of incidence either parallel(y axis) or perpendicular (x axis) the ripples.Therefore, we suppose the plane of incidence parallel to the xz plane, as in fig. 5.8,correspondingtotheperpendicular(⊥)geometry; thecalculationsinparallel(||)geometryare exactly the same, except swapping the axes x and y.We consider an EM plane wave travelling in the xz plane and approaching the surfaceat an angle of incidence θ. When the electric field is s-polarized, it always oscillates along
5.2. 2-DIMENSIONAL ARRAYS OF GOLD NANOPARTICLES 85plane ofincidencespzθyxFigure 5.8: Sketch of the reflection geometry, highlighting the differences between the spcoordinate systems, employed for the Fresnel coefficients (1.23), and the xyz principalaxes of the effective dielectric tensor ε eff .the y direction, independent of θ, so the wave propagates according to solely the dielectricfunction ε yyeff . The (complex) refractive index ñs effis then simply given by√ñ s eff = ε yyeff(5.15a)On the contrary, for p-polarized light the electric field lies in the xz plane, therefore thep wave “feels” the combined effects of the different dielectric functions ε xxeffand εzzeff , withrelative weights determined by the angle θ. ñ p effcan be deduced from the refractive indexellipsoid and writes [172, 212, 213]√ñ p eff = ñp eff (θ) = ε xxeff εzz eff +(εzz eff −εxx eff )sin2 θε zz(5.15b)effThe reflection and transmission of light can now be computed at any incidence angleand state of polarization of incoming light by applying eqs. (1.26) to the configuration offig. 5.5(c), and using ñ p,seff for the film and ñ s = √ ε s for the substrate.5.2.2 Optical anisotropy of self-organized Au nanoparticles arraysWith a theoretical description framework available, we can now successfully address theplasmonicpropertiesofthearraysofgoldNPsandseparatelyinvestigate thecontributionsof the NPs shape and of the mutual interactions to the collective optical anisotropy of thesystems. In §4.2.3 we already discussed qualitatively the influence of the NPs aspect ratioand spacings; here, we apply the model developed in §5.2.1 to quantitatively replicate theoptical response of the arrays and estimate the effects of the morphological characteristicson the LSP resonances.The calculations are performed employing the morphological parameters deduced byAFM analysis as geometrical input parameters for the model, in order to prevent theachievement of incidental agreement between data and model with non realistic samplecharacteristics.The first set of morphological constraints that we apply concerns the thickness of theeffective medium layer and the positioning of the inclusions. Assuming an ideal ridge-andvalleystructure, with regularly spaced facets tilted of 45 ◦ with respect to the substrate,
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Università degli studi di GenovaFa
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4 CONTENTS5.2.2 Optical anisotropy
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6 Introductionconfining the EM ener
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8 Introduction
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10 CHAPTER 1. THEORYEλBkFigure 1.1
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12 CHAPTER 1. THEORYfrom which, sep
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14 CHAPTER 1. THEORY≈ ω 0 like
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16 CHAPTER 1. THEORYε 1 (λ) = N 2
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18 CHAPTER 1. THEORYThe correspondi
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20 CHAPTER 1. THEORYso the induced
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22 CHAPTER 1. THEORYε MG −ε aε
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24 CHAPTER 1. THEORYUV range, this
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26 CHAPTER 1. THEORYwhich is called
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28 CHAPTER 1. THEORYwhere Γ 0 = v
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30 CHAPTER 1. THEORYpoints of the c
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32 CHAPTER 1. THEORYand then monoto
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Page 40 and 41: 40 CHAPTER 2. EXPERIMENTAL METHODSt
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Page 46 and 47: 46 CHAPTER 2. EXPERIMENTAL METHODSs
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Page 102 and 103: 102 CHAPTER 6. COMPOSITE MEDIA BASE
Page 104 and 105: 104 Conclusionsparticular cases of
Page 106 and 107: 106 Acknowledgements
Page 108 and 109: 108 BIBLIOGRAPHY[14] Shouheng Sun,
Page 110 and 111: 110 BIBLIOGRAPHY[44] Q A Pankhurst,
Page 112 and 113: 112 BIBLIOGRAPHY[74] Prashant K. Ja
Page 114 and 115: 114 BIBLIOGRAPHY[103] Tobias Kraus,
Page 116 and 117: 116 BIBLIOGRAPHY[137] F. Brouers, J
Page 118 and 119: 118 BIBLIOGRAPHY[169] Christy L. Ha
Page 120 and 121: 120 BIBLIOGRAPHY[201] G. Baldacchin
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