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

20 CHAPTER 1. THEORYso the induced polarization; moreover, these effects depend on the shape and relative sizeof the microscopic structures.Despite the extremely complicated microstructure of such composite media, in mostcases it is still possible to describe the macroscopic response of such a heterogeneousmaterial with an effective dielectric constant, that is an appropriate functional that “effectively”accounts for most of the optical characteristics of the specimens. In many cases,the effective dielectric constants of a mixture can be expressed in terms of a functional ofthe dielectric constants of each of the materials that enter the composite medium. Suchapproaches go under the name of “effective medium theories”. There are several methodsto derive effective medium theory; here, we start from the Clausius-Mossotti (CM)problem and generalize the solution to obtain the Lorentz-Lorenz, Maxwell-Garnett andBruggeman expressions [127, 128].The CM problem applies to a simple cubic lattice of polarizable points, with polarizabilityα and lattice constant a. When a uniform electric field E is applied, an electricdipole p = ε 0 αE loc is induced at each lattice point R n , which in turn generates an electricfield; the local field E loc is therefore determined by the superimposition of the externalfield and the fields E dip from the other dipoles:E loc (r) = E+ ∑ R nE dip (r−R n ) (1.28)whereE dip (r) = 14πε 03(p·r)r−r 2 pr 5 (1.29)The macroscopic polarization P is defined as the average dipole moment per unit volume,and in this case readsP = N V ε 0αE loc = nε 0 αE loc (1.30)where n = a −3 is the volume density of points. In order to write E loc as a function ofP and E we need to evaluate the sum in (1.28). This can be done using the Lorentzcavity method. In the simplest approximation we consider a sphere of radius ρ centeredin r: we explicitly add the dipole fields from the points inside the sphere and averagethe dipoles outside. Given the cubic symmetry of the lattice, the former contributionvanishes, while the latter equals to the volume integral of a dipole P, which is P/3ε 0[129]. Then, eq. (1.30) becomesand we obtain for P(P = nε 0 α E+ P )3ε 0Finally, applying the definition of the dielectric constantwe found the CM relation(1.31)nαP = ε 01−nα/3 E (1.32)D = ε o εE = ε 0 E+P (1.33)