## Abstract

We study the optical properties of metamaterials formed by layers of metallic nanoparticles. The effective optical constants of these materials are retrieved from the calculated angle-dependent Fresnel reflection coefficients for s and p incident-light polarization. We investigate the degree of anisotropy in the effective permittivity as a function of inter-layer spacing, particle size, filling fraction of the metal, and particle shape. For layers of spherical particles periodically arranged in a hexagonal lattice, the anisotropy disappears for the three inter-layer spacings corresponding to simple cubic (sc), bcc, and fcc volume symmetry. For non-spherical particles, an isotropic response can be still obtained with other values of the inter-layer spacing. Finally, we provide a quantitative answer to the question of how many layers are needed to form an effectively homogeneous metamaterial slab. Surprisingly, only one layer can be enough, except in the spectral range close to the particle plasmon resonances.

©2009 Optical Society of America

## 1. Introduction

Naturally occurring materials offer a limited range of optical properties that have been recently expanded by metamaterials. These are artificial materials that are textured on a small scale compared to the light wavelength, thus acting as homogeneous media of effective permittivity and permeability depending on their composition and microscopic structure. Extraordinary phenomena such as high-frequencymagnetism [1], negative-index behavior [2, 3], and the possibility of a perfect lens based on these concepts [4] have piqued the imagination of scientists to the point that metamaterials are sometimes identified with negative refractionmedia. This community has also gained momentum from recent advances in transformation optics [5], leading to concepts such as metamaterial optical cloaking [6, 7]. We are however interested in a more conventional, yet technologically interesting approach to metamaterials capable of displaying interesting electric response (generally without magnetic response), making use of nanoparticles of tens of nanometers in size as building blocks to synthesize larger-scale structures for controlling visible and near-infrared (vis-NIR) light with a wavelength that is >10 times larger than the particles [8]. The latter can be deposited in layers by self-assembly, generally leading to close-packed structures of the sort considered in this work [9, 10, 11, 12, 13, 14]. Similar structures have also been extensively studied in the search of photonic band gap materials [15, 16, 17, 18, 19].

The problem of obtaining the effective optical properties of composite materials has been addressed for a long time. The Clausius-Mossotti (CM) formula [20] relates the polarizability of small particles to the effective permittivity of an aggregate with either inversion symmetry or full disorder. This formula gives a reasonable description of the response of small-particle arrays, described through their dipolar response, and it is the basis of the Maxwell-Garnett (MG) theory [21] for obtaining the effective permittivity of a matrix material containing spherical inclusions of another material, with the permittivities of these media and the filling fraction of the inclusions *f* as the only parameters. When the shape of the constituent materials is not spherical, Bruggeman’s theory [22] can yield reasonable results in some cases, although it is generally better to use MG theory for spherical objects. However, both of these models fail at large filling fractions [23, 24]. More sophisticated theories have been presented, based upon spectral representations [25, 26, 27, 28]. The extension to higher multipoles beyond the dipoles leads to quasi-analytical results that have been worked out for several kinds of cubic lattices [29, 30, 31]. The effect of particle shape has also been investigated [32, 33, 34]. Other developments include extensions of MG theory to chiral mixtures [35], analysis of randomly oriented anisotropic particles [36], and nonlocal effects resulting from the microscopic granularity in composite materials [37].

In this work, we solveMaxwell’s equations rigorously for metamaterials formed by layers of metallic nanoparticles periodically disposed in a hostmedium. A scheme of the system is shown in Fig. 1. We discuss both spherical (isotropic) and ellipsoidal (anisotropic) nanoparticles. The difference in shape affects the degree of anisotropy of themetamaterial response. We investigate the optical response of these materials by first calculating their reflection coefficients, which are then compared to those of homogeneous anisotropic semi-infinite media, thus leading to the determination of effective optical constants.

We calculate the Fresnel reflection coefficients *R _{s}* and

*R*for s- and p-polarized light using a rigorous layer-KKR method [38, 39]. In this method, an incident plane wave is expanded in spherical waves centered around the particles, and the scattered outgoing spherical waves are re-expressed in terms of plane waves that describe the propagation of the electromagnetic field between consecutive particle planes. The reflection coefficients result from the self-consistent solution for the amplitude of the spherical and plane waves involved. We use a finite number of multiples with orbital momentumnumber

_{p}*l*≤

*l*

_{max}. The results presented below have converged for

*l*

_{max}=12. The number of plane waves that mediate the inter-layer interaction is also limited to a finite set around the specular reflection direction, with good convergence achieved for ~300 waves in the calculations here reported. This method has been previously used to study isotropic nanoparticle-based metamaterials [40].

The values of *R _{s}* and

*R*obtained by means of the layer-KKR method are then compared to those of a homogeneous anisotropic semi-infinite material, which we obtain from a straightforward generalization of the methods of Ref. [20]:

_{p}and

where

*ε*
^{eff} and *µ*
^{eff} are the electric permittivity and the magnetic permeability of the homogenous medium, the symbols ‖ and ⊥ refer to the response to field components parallel and perpendicular to the surface of the material (i.e., to the particle planes), respectively, and *θ* is the angle of incidence. By construction, the material response is isotropic with respect to the surface plane directions.

Values of *α, β*, and *γ* are obtained by comparing Eqs. (1) and (2) to the numerical results of the layer-KKR method for several angles of incidence. Although no magnetic response is expected in the system under consideration for small particle size, we show below that inclusion of an effective *µ*
^{eff} is important to describe retardation effects due to the finite size of the particles compared to the wavelength. In what follows, tabulated complex, frequency-dependent dielectric functions of gold, silver [41], and silica [42] have been used to describe the materials involved in these structures. Previous studies have focused in the transmission of finite metamaterial slabs to retrieve the effective optical constants [43].

## 2. Anisotropy due to lattice structure

We first consider systems formed by two-dimensional hexagonal arrays of spherical silver nanoparticles and discuss the anisotropy produced by non-cubic lattices. In order to avoid retardation effects in this preliminary discussion, the nearest-neighbor distance in each layer has been chosen *d*=6 nm in the calculations that follow, and we neglect nonlocal effects, which should be significant for these small dimensions [44, 45]. We also keep the filling fraction *f* (i.e., the metal volume fraction) constant. This is done by simultaneously modifying the radius of the spheres *r* (and thus their volume) and the distance between layers *d _{z}* (see Fig. 1). The effective dielectric constant is then calculated for fixed

*f*and for a range

*d*values.

_{z}For three specific different values of inter-layer distance *d _{z}*, 3D cubic lattices are obtained, with the layers under consideration oriented perpendicular to the 〈111〉 direction. More precisely, for

*d*=

_{z}*d*/2√6,

*d*=

_{z}*d*/√6, and

*d*=2

_{z}*d*/√6 we obtain a body-centered cubic lattice (bcc), a simple cubic lattice (sc), and a face-centered cubic lattice (fcc). In these particular instances, the system becomes isotropic. For other inter-layer distances, the response of the system is anisotropic.

The response of sc and fcc lattices is represented in Fig. 2 for a filling fraction of 30% within a range of wavelengths containing the dominant plasmonic response of the system. The actual quantity shown in the figure is Im{-1/e *ε*
^{eff}}, the so-called loss function, which can be directly measured using electron energy-loss spectroscopy [46] and exhibits a pronounced peak near the vanishing of the real part of the dielectric function (Re{*ε*
^{eff}}=0), signaling the existence of a bulk plasmon resonance. The loss function is isotropic for sc and fcc lattices, but it is anisotropic for values of *d _{z}* corresponding to non-cubic lattices. This is the case of the

*d*=1.31 lattice considered in Fig. 2, in which the degeneracy of bulk plasmons is broken for directions parallel and perpendicular to the layers (the difference in plasmon wavelength is ~10nm).

_{z}/dWe have calculated spectra for a range of *d _{z}* values and represented the plasmon resonance wavelengths as a function of

*d*for both field orientations in Fig. 3. The values of

_{z}/d*d*corresponding to the three cubic lattices are marked by vertical dashed lines. In these three cases, both the parallel and perpendicular components of the dielectric tensor exhibit plasmon resonance peaks at the same wavelength. Plasmons associated to parallel or perpendicular polarization with respect to the layer planes have different wavelengths except for these three crossing points of cubic symmetry. Two different filling fractions of the metal are considered in Fig. 3 (30% and 40%). The behavior is similar in both cases, but the higher filling fraction leads to lower plasmon wavelengths, which is due to stronger inter-particle interaction [40]. These conclusions are valid both for silver particles in vacuum and for gold particles in silica, although this latter system is of course more realistic.

_{z}/d## 3. Anisotropy due to particle shape

We have so far studied anisotropy in spherical-particle arrays due to a departure from cubiclattice symmetry. Anisotropy can also result from changes in particle shape even for cubic lattices. In this section, we study ellipsoidal particles, which unlike the spheres, cannot be handled analytically, and therefore, we have used the highly convergent boundary element method (BEM) [47, 48] to obtain the scattering matrix of the particles, and these are in turn introduced in the layer-KKR method described above to calculate effective permittivities.

We represent in Fig. 4 the loss function Im{-1/*ε*
^{eff}} for layered materials as a function of the aspect ratio of the particles *r*
_{⊥}/*r*
_{‖}, where *r*
_{‖} and *r*
_{⊥} are the ellipsoids semi-axes along directions parallel and perpendicular to the particle layers, and the ellipsoids are taken to be axially symmetric with respect to the layer normal direction. The geometrical parameters are varied in such a way that the metal filling fraction is kept constant. Three different values of *d _{z}/d* are considered, leading to two different plasmon resonances for electric-field polarization either parallel or perpendicular to the layers. However, these plasmons are degenerate for specific values of the ellipsoids aspect ratio that depend on

*d*. The fcc lattice predicts full degeneracy for spheres (

_{z}/d*r*

_{⊥}=

*r*

_{‖}), but isotropic behavior is observed in non-cubic lattices for oblate ellipsoids (

*r*

_{⊥}<

*r*

_{‖}) at short inter-layer spacings and for prolate ellipsoids (

*r*

_{⊥}>

*r*

_{‖}) in the opposite case. The anisotropy due to the lattice is compensated by the anisotropy produced by the particle shape in the crossing points at which degeneracy occurs.

## 4. Limits to homogenization

We have focused in the previous sections on the anisotropic optical properties of layered metamaterials and we have discussed the dielectric function neglecting the magnetic permeability, which is expected to be trivially *µ*
^{eff}=1 for the particle dimensions and material properties under consideration. Therefore, we expect to obtain *γ*=1 from our simulations [see Eq. (5)]. Furthermore, we also expect that cubic lattices yield *α*
^{2}
*β/γ*=1 [see Eqs. (3), (4), and (5)]. However, these conditions break down for relatively large particles, so that the effective-homogeneous-mediumassumption is no longer valid.

We put these conditions to a test in Fig. 5, in which the effective dielectric function (left panel) and the quantity *α*
^{2}
*β/γ*-1 (right panel) are represented for silver spheres of different size (*r*=5×10^{-4} nm, *r*=5nm, and *r*=20nm), arranged in a fcc lattice with a metal filling fraction of 40%. We expect this system to exhibit isotropic response, which it certainly gives outside the plasmon resonances (in particular, for wavelengths below 350 nm) or for very small particles (electrostatic limit). However, there is a significant departure from isotropy even for *r*=5nm particles at wavelengths around 400 nm (plasmon region). These anomalies are matched by large deviations from unity of the quantity *α*
^{2}
*β/γ*-1 in the plasmon region for all particle sizes, and over all the spectral region under consideration for *r*=20nm particles. Thus, in contrast to what can be expected, we conclude that even relatively small particles (e.g., 40 nm particles at a wavelength of 400 nm) are problematic when trying to define a homogeneous medium picture. The electrostatic effective dielectric function (obtained in the small-particle limit) yields a response that can be safely used for small particles (e.g., the 10 nm particles considered in next section), at least when calculating the reflectance and transmittance of finite particle films. However, the actual Fresnel coefficients, including their phase information, becomes problematic even at small particle sizes. Finally, it should be mentioned that Eqs. (1) and (2) work extremely well compared to the reflection coefficients calculated from the layer-KRR approach for a (wavelength-dependent) single choice of parameters *α, β*, and *γ* valid for all angles of incidence, although these coefficients do not satisfy the expected identity α2β/γ=1. Besides the problems noted with homogenization, there is also the possibility that the surface of our metamaterials does require a more careful description than that provided by the abrupt termination of an effective homogeneous film.

## 5. How many layers are needed to form a metamaterial

The concept of an effective permittivity is useful to describe the optical response of a metamaterial molded into arbitrary shapes, rather than having to deal with the detailed arrangement of particles that form it. Now, the question arises, what is the minimum size for which a piece of metamaterial still behaves as an effective homogeneous medium tailored into the same shape. We address this question here by studying the reflectance and transmittance of films formed by just a few particle layers.

In Figures 6 and 7 we show the normal-incidence reflectance and transmittance of films formed by 1, 2, 4, and 2^{25} (opaque film) particle layers (solid curves) arranged in a close-packed (fcc) structure with metal filling fraction of 40%. We have considered two different sizes of the spheres (radius *r*=5nm and *r*=20nm). We compare these results with the reflectance and transmittance of homogeneous films made of an effective material (broken curves). The effective dielectric function of the homogeneous material is calculated as explained above for a semi-infinite metamaterial of the same metal filling fraction in the electrostatic limit (*r*→0). The thickness of the homogeneous film is taken to be the number of layers times the inter-layer spacing *d _{z}* (even for a single layer, the value of

*d*corresponding to the fcc(111) structure is the appropriate one; we have verified this by calculating the reflectance for

_{z}*f*=10% (not shown), in which case

*r*≪

*d*, so that the actual choice of thickness in the effective homogeneous film is not so obvious). Interestingly, the homogeneous-film model works rather well even for a single layer, although there are some discrepancies within the spectral region of the plasmon resonances, and the model does not work above ~400nm for spheres of radius

_{z}*r*=20nm. This was expected, since we are using the electrostatic effective dielectric function for the homogeneousmedium, so that retardation effects become already important for 40 nm particles, as discussed in Sec. 4.

Further support to these results in provided by Fig. 8, in which the reflectance is studied as a function of incidence angle for the same systems as in Figs. 6 and 7, with particle radius *r*=5nm and fixed wavelength *λ*=360nm. The effective-homogeneous-film model works extremely well down to the single layer over a wide range of angles of incidence.

## 6. Conclusion

We have addressed the anisotropic optical response of multilayered metamaterials made of nanoparticles, in which the structure of the lattice and the shape of the particles can produce anisotropy. Interestingly, these two sources can cancel each other, resulting in specific configurations of non-spherical particles arranged in non-cubic lattices but yielding isotropic optical properties.

We have obtained significant deviations in the actual response of particle arrays with respect to the electrostatic limit for particles as small as 10–20 nm at visible wavelengths. Therefore, we conclude that one has to be careful when homogenizing the response of a composite system, as particle-size-dependent effects can be important even for small sizes compared to the wavelength, particularly near the plasmon resonances.

Surprisingly, a single layer of nanoparticles can behave like a homogeneousmaterial with the effective permittivity of an infinite metamaterial formed by layers arranged in a close-packed structure. This result is useful to assess the conditions under which a nanoparticle-based metamaterial can be assimilated to an effective homogeneous medium. Further work is still needed to explore the validity of the effective-homogeneous-medium assumption for metamaterials tailored into other non-planar shapes.

## Acknowledgments

This work has been supported by the Spanish MICINN (MAT2007-66050 and Consolider NanoLight.es) and by the EU (NMP4-2006-016881-SPANS and NMP4-SL-2008-213669-ENSEMBLE).

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