Interfacial Susceptibilities in Nanoplasmonics via Inversion of Fresnel Coefficients

Abstract

The reflection coefficients of a nanoparticle film are driven to a large extent by perpendicular and parallel interfacial susceptibilities that have the meaning of “dielectric thicknesses” which combine the actual geometry of the film and its dielectric properties. The direct determination of these parameters faces the long-standing issue of the derivation of complex optical constants from Fresnel coefficients via a unique spectroscopic measurement. The present work sets up an iterative algorithm based on inversion of the reflection coefficients recorded in the UV–visible range for two polarization states s and p and Kramers–Kronig (KK) analysis. To calculate the KK integrals over a limited energy window, the strategy was to complement measurements by spectra calculated in the framework of the spheroidal dipole approximation. The algorithm has been successfully tested on synthetic data of differential reflectivity for supported truncated spheres. These were chosen to span different dielectric behaviors, involving (a) for the particles, metals whose optical response is dominated by plasmonic excitations with a noticeable Drude behavior (Ag and Au) and (b) for the substrate, either nonabsorbing wide bandgap (alumina) or semiconducting (zincite and titania) oxides. Unlike the thin plate model, the approach was proven to apply to “dielectric thicknesses” of several tens of nanometres in cases in which, even for geometric sizes of the order of the nanometer, the classical long-wavelength dielectric approximation fails because of strong plasmon resonances. Therefore, the disentanglement of dielectric behaviors along the parallel and perpendicular directions simplifies the understanding on the interface polarization process by removing substrate contribution. The present work that deals with plasmonics in nanoparticles can be easily generalized to different morphologies as well as to other combinations of Fresnel coefficients.

Appendix: Supported Spheroid Polarizability in the Spheroidal Dipole Approximation

The aim of this appendix is to recall the expressions of the supported spheroid depolarization factors which are used as starting point of the inversion algorithm of SDRS. The geometry will be restricted to an oblate spheroid as it better describes the experimental particle flattening. Electrostatic interactions, both with the substrate and between particles, are kept at dipolar order for the sake of simplicity. Particles are ordered on a lattice of parameter L. Anyhow, equations for prolate case and up to the quadrupolar order can be easily found in the literature [3]. All the geometrical parameters are defined in the text and in Fig. 1.

The derivation of spheroid polarizabilities was obtained by Bobbert and Vlieger [3, 48, 56]; the Laplace equation is solved by using a multipolar spheroidal expansion [63] of the potential located both at the centers $\mathcal {O}$ and at the image $\mathcal {O}_{r}$ points in the substrate for all the spheroids. The continuity of the potential and of the normal derivative of the displacement field at the surface of the spheroids leads to an infinite set of linear equations for the spheroidal multipole coefficients. Once restricted to dipole interaction, the particle polarizability reads as follows:

$$\begin{array}{@{}rcl@{}} \alpha_{S,z} &=& \epsilon_{1} V \frac{\epsilon - \epsilon_{1}}{\epsilon_{1} + L_{z}(\epsilon - \epsilon_{1})} \\ \alpha_{S,\parallel} &=& \epsilon_{1} V \frac{\epsilon - \epsilon_{1}}{\epsilon_{1} + L_{\parallel}(\epsilon - \epsilon_{1})} \end{array} $$

(10)

where $V=4/3 \pi a^{3} \xi_{0}(1+\xi_{0}^{2})$ is the spheroid volume. The depolarizations factors are given by

$$\begin{array}{@{}rcl@{}} L_{z} &=& (1+\xi_{0}^{2})\left\{ 1- \xi_{0} \arctan\left(\frac{1}{\xi_{0}}\right) - \left(\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}\right) \xi_{0} \Xi(\xi_{1}) - \frac{V}{L^{3}(1 + \xi_{0}^{2})} \Sigma_{20}^{-} \right\} \\ L_{\parallel} &=& \frac{1}{2}(1+\xi_{0}^{2})\left\{ \xi_{0} \arctan\left(\frac{1}{\xi_{0}}\right) -\frac{\xi_{0}^{2}}{1+\xi_{0}^{2}} - \left(\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}\right) \xi_{0} \Xi(\xi_{1}) + \frac{V}{L^{3}(1 + \xi_{0}^{2})} \Sigma_{20}^{+}\right\} \\ \Xi(\xi_{1}) &=& \left( \frac{3}{2} + \xi_{1}^{2}\right) \xi_{1} \ln\left(1 + \frac{1}{\xi_{1}^{2}}\right) - \arctan\left(\frac{1}{\xi_{1}}\right) -\xi_{1} \\ \Sigma_{20}^{\pm} &=& \frac{1}{\sqrt{5 \pi}} \left[ S_{20} \pm \left(\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}\right) \widetilde{S}_{20}^{r}\right] \end{array} $$

(11)

The dipole–dipole interaction between particles comes into play through the lattice sums ($S_{20}$, $S_{20}^{r}$) over the direct $\mathbf {R}_{i}$ and image $\mathbf {R}_{i}^{r}$ points. They are defined through the following:

$$\begin{array}{@{}rcl@{}} S_{20}{}&=&{}\sum\limits_{i\ne 0} \left(\frac{L}{r}\right)^{3} \left.Y_{2}^{0}(\theta,\phi)\right|_{\mathbf{r}\,=\,\mathbf{R}_{i}} = - \frac{1}{4} \sqrt{\frac{5}{\pi}} \sum\limits_{i\ne 0} \left(\frac{L}{R_{i}}\right)^{3} \notag\\ \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} S_{20}^{r} {} &=& {} \sum\limits_{i\ne 0} \left(\frac{L}{r}\right)^{3} \left.Y_{2}^{0}(\theta,\phi)\right|_{\mathbf{r}=\mathbf{R}_{i}^{r}} = \frac{1}{4} \sqrt{\frac{5}{\pi}} \sum\limits_{i\ne 0} \left(\frac{L}{R_{i}^{r}}\right)^{3} \notag\\ &&{\kern9.5pc} \times\left(3 \cos^{2} \theta_{i}^{r}-1\right) \end{array} $$

(13)

where $Y_{2}^{0}(\theta ,\phi )$ is the spherical harmonics of order $\ell =2,m=0$. They depend on the spherical angle from direct $(\theta ,\phi )$ or image $(\theta ^{r},\phi ^{r})$ points (Fig. 1). The sums are calculated using the convergence trick of ref. [64]. The IS in the spheroidal dipole approximation are related to the above polarizabilities through the following:

$$ \gamma = \rho \alpha_{S,\parallel}, \quad \beta = \rho \alpha_{S,z} $$

(14)

where $\rho$ is the number of particles per surface unit.