The properties of any material depend on its constituents and their spatial arrangement. In the case of a random structure, it is impractical to explicitly account for all interactions within its volume due to the complexity of the resulting equations as well as, and in practice more importantly, a lack of transference of the properties to other realizations of such a material with different microscopic internals. Hence, what is needed is a description of macroscopic properties which, at this higher level, skip over the exact internal nanostructuring and focus on global characteristics, such as material properties and fractional amounts of constituent materials, stochastic description of their distributions, etc. These few parameters, which characterize the heterogeneous material on a macroscopic level, are then used derive a homogenized, simpler macroscopic description of the properties of the ensemble [19].
Maxwell Garnett Background
A macroscopic description of the effective electromagnetic properties of a heterogeneous medium involves creation of a permittivity function, which correctly accounts for how individual elements of the conglomerate interact with the external field. This implies that what is averaged is not the permittivities, but the fields and this requires careful accounting of the polarizabilities (α) of inclusions in the host medium. Assuming no interactions between inclusions, the effective permittivity 𝜖eff is given as [20]
$$ \frac{\epsilon_{\text{eff}}-\epsilon_{h}}{\epsilon_{\text{eff}}+ 2\epsilon_{h}}=\frac{4 \pi}{3}\sum\limits_{i} n_{i} \alpha_{i}, $$
(1)
where αi is the polarizability of inclusion i and ni is the particle density of species i. If we assume only one type of spherical inclusions, we then have in the quasistatic approximation that the polarizability α is expressed by the permittivity of host 𝜖h and inclusion 𝜖i media and its radius r (Fig. 1). For simplicity, one typically introduces the corresponding volume fraction of the inclusions as δ ≡ 4πnr3/3 [20],
$$ \frac{4 \pi}{3} n \alpha=\frac{\epsilon_{i}-\epsilon_{h}}{\epsilon_{i}+ 2\epsilon_{h}}\frac{4 \pi}{3} \mathit{n r}^{3}=\frac{\epsilon_{i}-\epsilon_{h}}{\epsilon_{i}+ 2\epsilon_{h}} \delta. $$
(2)
Combining the above with Eq. 1, yields the closed form of the Maxwell Garnett mixing formula
$$ \epsilon_{\text{eff}}^{\text{MG}}=\epsilon_{h}+ 3\delta\epsilon_{h}\frac{\epsilon_{i}-\epsilon_{h}}{\epsilon_{i}+ 2\epsilon_{h}-f(\epsilon_{i}-\epsilon_{h})}. $$
(3)
The MG formula utilizes only three parameters—the two permittivities and the volume fraction—to describe the macroscopic properties. While neither multiple scattering nor retardation are taken into account, these dependencies may be incorporated if needed. Hence, several extensions to the MG mixing formula have been proposed to account for various discrepancies introduced by using the effective medium approach [24, 42]. In these formulations, it is typically assumed that inclusions are distributed isotropically (on average) throughout the host medium and that they are exposed to a uniform field. However, this is not always the case, and the simple MG approach may fail when these conditions are not met, the problem we address in our work.
Gradient effective permittivity (GEM) model
Here, we consider situations outside the limits of applicability of the classical MG formula, that is when anisotropically distributed inclusions within a host medium are illuminated by an inhomogeneous electric field. A typical example, where such conditions occur are thin nanoparticle layers supported by interfaces [12, 15, 43], where the presence of the interface breaks symmetry and introduces electric field gradients (see Fig. 1a). These can be generated by, e.g., reflection and interference on a dielectric substrate or the excitation of a surface plasmon polariton if the supporting material is metal.
In our proposed gradient effective medium approach (the GEM model), instead of homogenizing a nanoparticle layer into a single effective one like in an MG approximation, the nanoparticle layer is divided into several segments (thin sublayers) of equal thickness and each segment is homogenized separately (see Fig. 1b,c) using the classical MG mixing formula. The motivation behind such treatment is the fact that a supported nanoparticle layer consists, in principle, of objects of various size, Fig. 1a. By itself, this would not constitute a problem; however, the nanoparticles are not distributed equally in the direction normal (e.g., z-axis) to the substrate. Instead, they are placed at the interface. Hence, the average mass distribution is given by a particular function of z with parameters dependent on the particle distribution, c.f. an exemplary volume fraction dependence in Fig. 2a. When coupled with an inhomogeneous near-field of an LSPR, this function becomes important and needs to be taken into account in an effective treatment, which is what we do presently. We assume that the nanoparticles are sufficiently small to be described by the quasistatic polarizability, and we omit radiative coupling. For further simplification, no explicit interactions between layers are assumed within the model.
The key parameter in the GEM model, like in the MG approach, is the volume fraction δ—the ratio between the volume of inclusions, nanoparticles on the substrate, and the total volume of the layer. The distinction between the models is that the volume fraction is different in each sublayer and depends on the distance from the substrate as well as the distribution of particles in the layer. We base our initial analysis of nanoparticle layers on previously reported Pd particle distributions [10, 11], which are therein characterized by a mean and standard deviation. For convenience in calculations, we assume a Gaussian distribution of Pd nanospheres around a given mean radius μr with corresponding standard deviations σr, although in reality a spherical shape may not be adequate. However, the general procedure is the same for any MG homogenized layer composed of individual nanoobjects, as in the GEM method the permittivity of each sublayer is based on the volume fraction of inclusions in that sublayer (and on the permittivities of the inclusions and the host medium).
The total thickness of the graded medium, which is equal to the largest assumed diameter given by H = 2(μr + 3σr), is divided into Np sublayers of thickness h = H/Np, see Fig. 1b. Each l th-sublayer contains a different amount of material V l (of inclusions), which is given by a sum (integral) of all sphere segments (volume \(V_{\text {seg},i}^{l}\)) contained within this sublayer-l (Fig. 1b) over the radius distribution. The volume fraction of a sublayer δl can then be written as
$$ \delta_{l}=\frac{V_{l}}{V_{\text{sub}}}=\frac{{\sum}_{i} V_{\text{seg},i}^{l} {N_{i}^{l}}}{\mathit{s h}}=\frac{N_{p}}{H}\sum\limits_{i} V_{\text{seg},i}^{l} \frac{{N_{i}^{l}}}{s}, $$
(4)
where i runs over all sphere segments in a given sublayer of volume V sub and Ni is the number of particles per layer area s. This allows us to write the expression for the effective permittivity of the l th-sublayer, see Fig. 1c,
$$ \epsilon_{\text{eff},l}=\epsilon_{h} \frac{2 \frac{N_{p}}{H} {\sum}_{i}{V^{l}_{\text{seg},i}\frac{{N^{l}_{i}}}{s}(\epsilon_{i}-\epsilon_{h})+\epsilon_{i}+ 2\epsilon_{h}}}{\epsilon_{i}+ 2\epsilon_{m}+\frac{N_{p}}{H} {\sum}_{i}{V^{l}_{\text{seg},i}\frac{{N^{l}_{i}}}{s}(\epsilon_{i}-\epsilon_{h})}}, $$
(5)
which is simply the MG formula applied independently to each slice of the homogenized graded medium. At this point and, indeed, throughout this work, we only consider a single material comprising the nanoparticles; however, the above equation can easily be expanded to include additional materials with different permittivities as is commonly reported in literature for the MG approximation [19]. This can be directly done by expanding the sums in Eq. 4 over index i to consider materials with different permittivities.
The volume fraction distribution, Fig. 2a, in each layer of nanoparticles, manifests itself in the effective dielectric function of the effective medium layer (see Fig. 2b, c). Both parts of the effective dielectric function increase with an increasing volume fraction, as plotted in Fig. 2. The sublayer with the largest volume fraction is the one which contains the middle segment of those spheres whose radii are close to the mean size μr. The permittivity decreases with increasing distance from that sublayer, although in an asymmetric manner because sublayers close to the substrate contain contributions from both small and large particles, while sublayers beyond μr contain contributions only from larger ones.
A large particle size dispersion leads to a more uniform effective permittivity distribution across the gradient layer (e.g., 8 ± 3 nm), whereas nanoparticle layers with narrow radius distributions (1.5 ± 0.3 nm) are homogenized into gradient media with pronounced permittivity variation—the effect of different size distributions on 𝜖eff is shown in Fig. 2b, c. As the response of any electromagnetic system depends on the overlap of material and field, the manner in which the homogenization procedure is carried out affects the end result. As electromagnetic fields are rarely homogeneous, we expect that the gradient effective medium model will improve the accuracy of simulations over the MG approach, especially when considering strongly inhomogeneous electric fields. This improvement will be especially apparent for various types of surface plasmon sensors, which offer greatly enhanced, but quickly decaying, fields near the metal-dielectric interface.
Numerical Calculations
In order to test the gradient model, we use interchangeably the transfer matrix method (TMM) and the FDTD method. The former are used in initial evaluation of discontinuous nanoparticle layers supported by flat substrates, in cases in which the nanoparticles are homogenized and the structure is planar. The latter are used in every case in which discrete nanoparticles are simulated as well as when computing the optical response of an LSPR sensor decorated with discrete or homogenized nanoparticles.
Modeled structures are placed on a substrate with a refractive index of 1.45. The parameters of discontinuous Pd nanoparticle layers are initially based on those fabricated for the work of Adibi et al. [10], but subsequently tested mean radii and their standard deviations are probed in a broad range, as can be observed in other experiments [12]. The initial equivalent mass thickness is assumed to be 0.5 nm [10]. In calculations, the permittivity of Pd, as well as of Ag when needed for the LSPR sensor, is taken from Palik [44], while for Au from Johnson and Christy [45]. FDTD calculations are carried out using FDTD Solutions from Lumerical, Inc., Canada. Due to the presence of nanoparticles with sizes down to 2 nm in diameter, a mesh resolution of 0.5 nm is used. As the plasmon resonance of such small Pd nanoparticles is far outside the considered wavelength range, the accuracy, which is enhanced by the use of subpixel averaging, is adequate.
In the case of nanoparticles supported by flat substrates, we use periodic boundary conditions in the transverse directions, while in simulations of nanoplasmonic sensing perfectly matched layer absorbing boundary conditions are used. The simulation area is 200 × 200 nm2. In simulations with discrete nanoparticles, the support surface is uniformly decorated with non-overlapping nanoparticles with randomly drawn positions. For FDTD, we generate a set of positions with radii being drawn from an appropriate Gaussian distribution with mean radius μr and its standard deviation σr. In TMM, we generate a vector of discrete radius values ranging from zero to three standard deviations. Then, the number of particles per area is calculated for each radius value based on the Gaussian distribution normalized so that the total particle volume corresponds to an equivalent mass thickness of 0.5 nm.