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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Plasmonic nanocomposites find many applications, such as nanometric coatings in emerging fields, such as optotronics, photovoltaics or integrated optics. To make use of their ability to affect light propagation in an unprecedented manner, plasmonic nanocomposites should consist of densely packed metallic nanoparticles. This causes a major challenge for their theoretical description, since the reliable assignment of effective optical properties with established effective medium theories is no longer possible. Established theories, e.g., the Maxwell-Garnett formalism, are only applicable for strongly diluted nanocomposites. This effective description, however, is a prerequisite to consider plasmonic nanocomposites in the design of optical devices. Here, we mitigate this problem and use full wave optical simulations to assign effective properties to plasmonic nanocomposites with filling fractions close to the percolation threshold. We show that these effective properties can be used to properly predict the optical action of functional devices that contain nanocomposites in their design. With this contribution we pave the way to consider plasmonic nanocomposites comparably to ordinary materials in the design of optical elements.

Plasmonic nanocomposites, as considered here, consist of metallic nanoparticles embedded in a dielectric host material [

If metallic nanoparticles are brought together with a sufficient density in a dielectric host, they constitute a referential example of what is nowadays known as a metamaterial [

Having such tailored materials at hand would significantly enlarge the degrees of freedom in designing optical devices. To be specific, nowadays, considering only naturally available materials, the optical properties of a specific optical element, such as a coating or a lens, have to be looked up in a database. Additionally, in this database, only a few discrete materials are available. This constitutes a severe limitation in the design of many optical devices. Plasmonic nanocomposites promise to overcome this limitation by their ability to provide materials with adjustable and tunable properties. However, to consider plasmonic nanocomposites in the design process of functional optical devices, it is of paramount importance to assign effective properties to these materials that can reliably predict the optical response.

The assignment of effective properties to plasmonic nanocomposites is a long standing problem and many solutions and associated theories were suggested [

Here, we solve this problem by assigning effective properties to dense plasmonic nanocomposites made from amorphously arranged metallic nanoparticles using full wave simulations in the first instance [

The unique feature of our analysis is the application of large-scale computational resources. This permits us to consider extended spatial domains of nanocomposites such that their intrinsic properties are entirely reflected. The basic geometry of the considered structure is shown in

Motivated by various experimental results, we consider plasmonic nanocomposites made from silver or gold nanoparticles here [_{2} matrix by a suitable co-sputtering technique. The mean diameter of the nanoparticles and their filling fraction can be extracted from high resolution transmission electron microscope (HRTEM) and energy dispersive X-ray spectroscopy measurements, respectively [_{2} was considered as a nondispersive material with a permittivity of 2.25 and the material properties of Si as considered in some parts of our work, are taken from the literature [

It should be noted that the nanocomposite as considered here is simplified in two ways compared to the real structure. First, it is assumed that the nanoparticles are monodisperse,

Second, a strictly spherical shape is assumed. This, of course, is not entirely correct but it is nevertheless a reasonable approximation. Due to the minimization of the surface energy, the nanoparticles always grow as spheres. We only observe deviations from spherical shape at higher filling fractions if two growing particles approach each other. Moreover, the consideration of more complicated geometries for each nanoparticle is cumbersome since only marginal information is available on their precise shape. However, we wish to stress that the different shapes will introduce a dispersion in the resonance frequencies of the polarizabilities of the individual nanoparticles [

For our simulations, the finite-difference time-domain (FDTD) method [

A selected snapshot of the amplitude of the electric field in a cross-section through a referential silver plasmonic nanocomposite is displayed in

To extract the complex reflection and transmission coefficients through a slab of the nanocomposite, the total electric field in a plane directly above or below was spatially averaged. Whereas this provides the amplitude of the transmission coefficient directly, the reflection coefficient was calculated by subtracting the incident field prior to spatial averaging. This incident field was obtained in a supporting simulation where the propagation of the illumination in the absence of the sample was simulated. It should be noted that the reflected and transmitted amplitudes are non-zero, within numerical precision, only for the electric field component of the linearly polarized illumination. No depolarization was observed. This strongly supports the claim of considering the plasmonic nanocomposite effectively as an isotropic material.

To assign effective properties to the plasmonic nanocomposite, we invert the complex reflection

Here, _{0} = 2_{0} is the wave number of the free space wavelength. These effective wave parameters, as they are usually called, are linked to the effective material parameters permittivity

To compare the obtained results to analytical predictions from traditional homogenization theories, we use a Maxwell-Garnett effective medium theory. There, the effective permittivity is calculated according to:

with

Referential results of the effective properties for a plasmonic nanocomposite made from silver nanoparticles are shown in

In both cases, a strong dispersion of the effective permittivity can be seen around a resonance wavelength close to 420 nm. This resonance wavelength is best extracted from the peak in the imaginary part of the effective permittivity. However, resonance strength, as well as line width strongly deviate among both methods. We would like to stress that the correct properties are those retrieved from the full wave simulations. Only these parameters correctly predict the actual reflection and transmission from a slab. The strength of the resonance of the true effective properties is approximately half the strength of the resonance predicted using the effective medium theory. Moreover, the line width of the true effective properties is equally enlarged by approximately a factor of two. An exact quantification, however, is not possible, since we clearly see in

Simultaneously with the much broader resonances, also, a spectral region of anomalous dispersion was encountered, where the real part of the permittivity increases with the wavelength. This is in contrast to an ordinary dispersion which suggests a decreasing permittivity with an increasing wavelength. We would like to stress that such an anomalous dispersion is a generic feature of all resonances and not unique to the material under consideration, although the spectral domain is extraordinarily large. Again, it is large, because the dispersion is caused not just by a single Lorentzian oscillator, but rather multiples thereof with slightly detuned resonance wavelengths.

From the quantitative difference of the effective properties as retrieved with the different methods we can conclude that the Maxwell-Garnett theory is not applicable anymore for these dense nanocomposites. Futuremore, an

The effective properties of silver nanocomposites depending on the filling fraction are shown in

Considering excessively large filling fractions,

The analysis discussed above for nanocomposites made from silver can be carried out with nanocomposites made from gold as well. Selected results in terms of effective properties for a nanocomposite with a 20% filling fraction are presented in

The most interesting feature of the analysis are the larger values in the achievable effective permittivity if compared to nanocomposites made from silver with the same filling fraction. The imaginary part, where those aspects are usually best perceived, is larger by 50%. This is surprising, since common wisdom usually suggests that silver is the preferable plasmonic material, because its intrinsic absorption is much less than that of gold. This implies a much better fulfillment of the Fröhlich condition, a larger value of the polarizability in resonance for otherwise identical systems and, hence, a much stronger dispersion. However, it also seems that the coupling to neighboring nanoparticles is enhanced. This is detrimental for the effective properties. The spread in resonance wavelengths of eigenmodes supported by the nanocomposite eventually limits the achievable maximal value for the permittivity. In contrast, if the particles are less vulnerable against such an interaction, since non-radiative losses dominate over radiative losses, the maximal achievable values for the dispersion of the nanocomposite is larger. This holds for the situation under consideration, since gold has a much larger imaginary part in its intrinsic permittivity than silver. In summary, neighboring nanoparticles do not experience each other in the gold nanocomposite, since they tend to absorb the incident light and not to scatter it. This is an important distinction that has to be considered in choosing a suitable material for a specific application. This peculiar feature would not be observable in an effective medium description based on Maxwell-Garnett theory, as can be also seen in the figure.

After all, the retrieval of effective properties is important. However, these effective properties are only useful if they can predict the optical properties of functional devices. Motivated by recent experiments, we consider here the use of a plasmonic nanocomposite made from gold as an integrated part of a bilayer anti-reflection structure on top of a silicon substrate. The effective properties of the considered nanocomposite are shown in

Results of the simulated reflection as a function of the wavelength for selected spacer thicknesses are displayed in

As a short conclusion, we have discussed the effective optical properties of plasmonic nanocomposites with an extremely large filling fraction. The filling fraction is in the region where traditional effective medium theories are no longer applicable. This requires the use of full wave optical simulations. The effective properties show anomalous features, such as the obvious appearance of multiple resonances in the spectrum despite the use of identical nanospheres. The strong horizontal coupling of metallic nanoparticles in the nanocluster has been identified as the reason for these features. We discussed the dependency of effective properties on the filling fraction and on the material from which the nanocomposites are made. Interestingly, nanocomposites made from gold have a stronger dispersion. This is against common wisdom suggesting that gold as a plasmonic material is usually inferior to silver. We have argued here that the stronger intrinsic absorption of gold eventually turns out to be something useful, since it suppresses the interaction between neighboring nanoparticles in the nanocomposite. This suggests that higher filling fractions are feasible without encountering detrimental effects. We wish to stress that similar insights have been obtained while discussing the optical properties of meta-atoms with an electric and magnetic dipole resonance in the transition from a periodic to an amorphous arrangement [

Finally, the effective properties of the nanocomposite have been shown to be useful for predicting the optical response of functional optical devices,

The work of Christoph Etrich, Stephan Fahr and Carsten Rockstuhl was supported by the German Federal Ministry of Education and Research (PhoNa) and by the Thuringian State Government (MeMa). Mehdi Keshavarz Hedayati, Franz Faupeland Mady Elbahri gratefully acknowledge the financial support by the German Research Foundation (DFG) through the projects EL 554/1-1 and SFB 677 (C1,C9). Mady Elbahri would like to thank the Initiative and Networking Fund of the Helmholtz Association (Grant No. VH-NG-523) for providing the financial base for the start-up of his research group. We thank Karsten Verch (

The authors declare no conflicts of interest.

_{2}nanocomposite thin films

Schematic of the configuration under consideration. A plasmonic nanocomposite made from densely, but randomly, arranged metallic nanoparticles shall be an integrated part of an optical device, e.g., as an anti-reflection coating. The consideration of the entire material in an optical analysis of such a device is too complicated. Therefore, it should be described by effective optical properties to predict the reflected and transmitted light properly; as indicated in the figure.

Amplitude of the electric field in a cross section through the plasmonic nanocomposite (between z = 0 nm and z = 20 nm). The incident linearly polarized plane wave propagates in the +z-direction. The plasmonic nanocomposite has a filling fraction of 20% and consists of randomly arranged non-touching silver nanospheres with a diameter of 4.1 nm embedded in a generic mondisperse dielectric material (

Effective properties of a plasmonic nanocomposite with a filling fraction of 20%. The effective properties are calculated by means of full wave finite-difference time-domain (FDTD) simulations (solid lines) and a Maxwell-Garnett effective medium theory (dashed line). The real part of the effective permittivity is shown in blue whereas the imaginary part is shown in red.

Rigorously calculated effective properties of a silver nanocomposite depending on the filling fraction. (

Effective properties of a gold nanocomposite with a filling fraction of 20%. The effective properties are calculated by means of full wave FDTD simulations (solid lines) and a Maxwell-Garnett effective medium theory (dashed line). The real part of the effective permittivity is shown in blue whereas the imaginary part is shown in red. The plasmonic nanocomposite consists of randomly arranged non-touching gold nanospheres with a diameter of 4.1 nm embedded in a generic nondispersive dielectric material (

(

Angle dependent reflection from a silicon substrate that is covered with a dielectric spacer (