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

24 CHAPTER 1. THEORYUV range, this approximation can adequately describe the optical response of sphericaland ellipsoidal particles with sizes below ≈100 nm.a zε ma ya xε hFigure 1.10: Sketch of a isolated metallic ellipsoidal particle, with principal semiaxis (a x ,a y , a z ) and dielectric function ε m , immersed in a dielectric host of dielectric constant ε h .Let’s consider a metallic ellipsoidal particle immersed in a transparent dielectric hostand far from any other polarizable entity (fig. 1.10). The ellipsoid has semiaxes a γ (γ =x,y,z) oriented along the cartesian axes, and the dielectric constants of the metal andthe host are, respectively, ε m and ε h (the latter purely real). We start by considering theeffectsofastaticappliedelectricfieldE 0 . Astheparticleisnotspherical, thepolarizabilityis a tensor α. Then, in general the induced electric dipole p is not parallel to E 0 [129, 148]:p = ε 0 α⊗E loc (1.40)where E loc is the local field acting on the particle, and differs from E 0 due to the polarizationof the host. If the host is homogeneous and isotropic, then E loc = ε h E 0 , and theprevious equation becomesp = ε 0 ε h α⊗E 0 (1.41)For ellipsoidal particles α is diagonal, and has principal values α γ given by [148]ε m −ε hα γ = vε h +L γ (ε m −ε h )γ = x,y,z (1.42)where v = 4π/3 a x a y a z is the particle’s volume and L γ are the depolarization factors,that can be written asL γ = a xa y a z2∫ ∞0dq(q +a 2 γ)√ ∏η=x,y,z (q +a2 η)(1.43)and satisfy the sum rule ∑ γ L γ = 1.For spherical particles (fig. 1.11(a)) we have a x = a y = a z ≡ a and the geometricalfactors reduce to L γ = 1/3 in all directions: the polarizability is isotropic and from (1.42)we findα sph = 3v ε m −ε hε m +2ε h(1.44)