Refined effective-medium model for the optical properties of nanoparticles coated with anisotropic molecules

This work aims to provide a simple yet complete effective dielectric function for an anisotropic layer of polarizable molecules adsorbed on a metallic surface. This effective medium model considers the important and nontrivial case of nonvacuum embedding media and accounts for orientation effects, coverage dependence through dipole-dipole interactions, and image-dipole effects. To check the model's validity, we focus in particular on the experimentally relevant case of dyes adsorbed on metallic nanospheres. We can then use anisotropic Mie theory, together with the effective dielectric function describing the molecular coating, to calculate their optical properties. We show that this effective medium description is in very good agreement with more elaborate and computationally intensive microscopic calculations based on coupled-dipole models. The effective medium model therefore provides a simple means to investigate orientation effects and coverage dependence, including in more complex systems such as dyes adsorbed on nonspherical or ensembles of nanoparticles. This model can readily be used to further our theoretical understanding of dye-nanoparticle systems, for example in the context of dye-plasmon resonance coupling or surface-enhanced Raman and fluorescence spectroscopy.

Figure 1

(a) Schematics of discrete polarizable dipoles adsorbed onto a nanoparticle at a distance $d$ from the surface. The elongated ellipsoids represent a uniaxial polarizability tensor, here with random orientation. (b) Corresponding effective medium model where the dipoles are replaced by a continuous shell of thickness $L$ and dielectric function ${\underline{\varepsilon }}_s$, possibly tensorial.

Figure 2

Shell-thickness ($L$) dependence of the differential absorption cross section for an isotropic EMM for (a) the dilute regime ($\rho =0.01 {\mathrm{nm}}^{-2}$) where ${\varepsilon }_s$ is defined by Eq. (4), and (b) high coverage ($\rho =0.6 {\mathrm{nm}}^{-2}$), using Eq. (6) instead. A strong $L$ dependence is observed in the latter case. The silver sphere radius is $a=14$ nm.

Figure 3

(a)–(c) Anisotropic-EMM predictions for the absorption spectrum of an $L=1$-nm-thick planar layer of perpendicular (a), in-plane isotropic (b), or fully isotropic (c) dipoles in water compared to the analytic solutions of a 2D array of dipole given in Appendix pp1 (dashed lines). The agreement is excellent. The isotropic-EMM model is also shown as a dotted line in (c). For spherical arrangement of molecules, the same anisotropic EMM is used, as shown schematically in (g). (d)–(f) The predictions for spherical shells in water are compared to microscopic coupled-dipole models (dashed line) for perpendicular (d), in-plane (e), and isotropic (f) arrangements. The inner radius of the water sphere is 14 nm. Different coverages $\rho$ are considered in all cases. Equations (17, 18, 19) are used for the anisotropic EMM with $d=1$ nm (but the image-dipole effect is negligible and could also be ignored in this case as there is no metal).

Figure 4

Shell-thickness ($L$) dependence of the differential absorption cross section for an anisotropic EMM of a spherical layer of perpendicular (top) and parallel (bottom) adsorbates in water (a),(d) or around a silver sphere (b),(e). The latter case shows an undesirable $L$ dependence, which can be avoided using an improved two-layer configuration (c),(f), where the effective shell layer is separated from the sphere surface by a distance $d-L/2$. In all cases, the core has radius $a=14$ nm and a coverage of $\rho =0.6 {\mathrm{nm}}^{-2}$ is used, but similar conclusions are obtained at other coverages. The results are compared to CDM and NP-CDM predictions for dipoles at a distance $d=1$ nm from the water (fictitious) or silver core.

Figure 5

Comparison between the EMM model and NP-CDM at low coverage $\rho =0.01 {\mathrm{nm}}^{-2}$ for perpendicular (a) and parallel (b) orientations, for varying dipole distance from the surface, $d$. We here use the two-layer model with a small $L=0.2$ nm.

Figure 6

Comparison between the two-layer EMM model and NP-CDM at a dipole distance where image-dipole effects are important, $d=0.5\mathrm{nm}$, as a function of coverage for perpendicular (a) and parallel (b) orientations, for varying dipole distance from the surface, $d$. We use $L=0.1$ nm and the thickness of the spacer water layer is $d-L/2$. In (c) the parallel EMM model [same as in (b)] is instead compared to NP-CDM for randomly oriented uniaxial dipoles.

Figure 7

Surface-averaged field intensity enhancements (perpendicular, left axis, and parallel, right axis, components) at a distance $d$ from a 14-nm-radius silver sphere in water. Note the different scale for the two $y$ axes.

Figure 8

Simulated absorption spectra for dipolar dimers with separation 0.9 nm. Two types of dipolar polarizabilities are compared: plane-isotropic (red curve), or uniaxial with in-plane random orientation (100 representative light-gray curves are shown). The relative orientation of both dipole moments strongly affects their coupling; the two extreme cases of head-to-tail and side-by-side configurations are presented as dashed gray curves. The average spectrum of any two dipoles randomly oriented in the plane is shown in blue. For reference, the absorption spectrum of a single isolated dye is shown by a black dashed line. All spectra are orientation averaged over all directions of incidence and polarizations.