1. Introduction

Nanocomposite metamaterials (MMs) with the effective permittivity close to zero (so-called epsilon-near-zero, or ENZ MMs) have attracted a lot of interest in recent years. Such materials, when properly designed, can potentially be utilized in multiple technologically important applications. Furtheremore, they can potentially advance a number of optical and electronic components. So far, most studies on the subject, both the experimental [1–3] and theoretical ones (see, e.g., [4] and references therein), have focused on narrowband ENZ MMs, when the condition Reε_eff ≈ 0 is fulfilled in the vicinity of one frequency only. At the same time, extended bandwidth may be a critical issue for many optical applications, such as, e.g., multi-color and hyperspectral imaging with high resolution [4, 5], cloaking devices [6], or perfect absorption [7]. Very recently, consideration has been given to broadband ENZ MMs [8–13], but the most papers were restricted to one-dimensional (1d) and quasi-1d nanocomposites. The only exception is the paper [10], where an assembly of spherical metal alloy particles embedded in a dielectric host has been suggested to null the real part of the effective permittivity over a frequency range. Thus, in spite of its importance, the problem of designing broadband 3d ENZ MMs has not been addressed so far. The present study was undertaken to partially fill this gap.

In the 3d case, dealing with accurate ab initio numerical simulation is prohibitively time and memory consuming. Because of this and also seeing the present study as a preliminary to a more detailed and complicated analysis, in what follows we restrict ourselves to consideration of very simple approach based on classical homogenization theory. The basics of this theory were laid down by Maxwell Garnett in 1904 [14]. Maxwell Garnett theory (MGT), based on the concept of the local Lorentz field, dealt with a dilute suspension of homogeneous spherical particles embedded in a host medium. Using a plane-wave-expansion method, Datta et al. obtained a good agreement of the MGT with their results for dielectric spheres arrayed in face-centered cubic, simple cubic, and diamond lattices [15]. For random suspensions of spherical particles, it was shown that the MGT predicts the effective response very well when the contrast ratio is not too high and the spheres are more or less separated [16]. The MGT withstands the most stringent test for superconducting inclusions (the infinite contrast ratio) for inclusion volume fraction up to 0.3 when compared with the strong-contrast expansion in terms of three-point correlation function [17]. Generally, for disordered systems, a red shift and broadening of absorption peaks occur as compared to the MGT; these effects are largest when the filling factor is about 10 – 20%.

We use more sophisticated homogenization technique dealing with spheroidal shapes. The necessity of such a complication stems from the fact that to design a composite with desired optical properties over a frequency band, it should have a sufficient number of degrees of freedom. This can be achieved by two means. Because the effective permittivity depends on both the material parameters (constituents’ permittivities) and geometrical parameters, we could change either latter or former. So, dealing with metal alloy spheres allows one to vary the particles’ composition and, hence, to change their polarizabilities and tune the effective permittivity [10]. Here, we propose to control the effective permittivity of the composite through designing its geometry. More specifically, we deal with such geometry as a dilute random suspension of metal nanoparticles of spheroidal shape. To have the nesessary number of degrees of freedom, polydisperse suspensions have been considered.

Formally, our technique can be used in the visible as well as in the IR spectral regions. However, its practical implementation can cause some technical issues. For example, one of the major issues with the implementation of the technique in the visible spectral range is that because our composite must be macroscopically homogeneous, many particles of different kind must be placed and randomly oriented inside a relatively small volume. At the same time, in the mid-IR and far-IR, our basic geometry includes extremely sharp needles and/or very flat disks. That is why the fabrication of such composites micht be a challenging task.

2. Nulling technique and basic approximations

In our consideration, the problem of broadband permittivity nulling on the band [ω₁, ω₂] can be formulated as finding a minimum of the function

We focus on geometry which allows an analytical solution for the effective permittivity. This solution may be considered as a Maxwell-Garnett type approximation. It applies to composites where some particles (inclusions) are embedded in a dielectric host (matrix). If the inclusions are of spheroidal shape (see Fig. 1), an analytical solution is available. As it is well known, the Maxwell-Garnett type approximation works the best for dilute suspensions, when the total filling factor of inclusions is much less than unity. On the other hand, if the total filling factor is too small, efficient nulling becomes hardly possible (rms becomes too large). Because of this, in what follows we fix an intermediate value of the total filling factor (f = 0.08) and consider rms as a function of the central wavelength λ₀ = 2πc/ω₀ for different values of the dimensionless bandwidth. In fact, the required permittivity nulling may also be accomplished for more concentrated suspensions of spheroidal particles. As an example of such homogenization technique we could use, e.g., asymmetric Bruggeman approximation for spheroids [18]. However, it is nesessary to keep in mind that an increase of the metal filling factor can result in undesired increase of the electromagnetic losses.

 

Fig. 1 Geometry of the composite under consideration: suspension of randomly oriented and variously shaped metal spheroids with the permittivity ε₁ embedded in a dielectric host with the permittivity ε₀.

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The homogenization of such an assembly was considered by Nan et al. in terms of the T-matrix formalism [19] and independently by Gao et al. [20,21] who considered nonlinear optical properties of metal-dielectric composites. Although they only considered spheroids of one kind (fixed shape), the generalization of their results to the case of variously shaped spheroids is straightforward. Indeed, within the framework of the MGT, the effective permittivity can be written as [20, 21]

In real nanoparticle suspensions, nanoparticles are usually shape-distributed. The question arizes how to accurately take into account their depolarization factor or aspect ratio distribution. In order to do that, we invoke earlier obtained results [23], properly adopted for polydisperse spheroid suspensions. In this case, the effective permittivity reads

When dealing with metal nanospheroids, it is necessary to keep in mind that if at least one of their dimensions is too small, one cannot use the bulk values of the metal permittivity (due to surface scattering, the limitation of the electron mean free path must be taken into account). If so, Eq. (6) must be modified in such a way that ${\epsilon }_1={\epsilon }_{1\Vert }^{(i)}$ when calculating ${\beta }_{\Vert }^{(i)}$, and ${\epsilon }_1={\epsilon }_{1\perp }^{(i)}$ when calculating ${\beta }_{\perp }^{(i)}$, where ${\epsilon }_{1\Vert }^{(i)}$ and ${\epsilon }_{1\perp }^{(i)}$ take into account corresponding corrections for surface scattering.

3. Numerical results

In what follows, we assume gold as the metal constituent in the composite. Its permittivity has been taken by using the Drude-critical points model which is shown to be rather accurate at wavelengths above 200 nm [24]. Teflon (ε_d = 1.823) has been considered as host medium (ε_d ≡ ε₀).

As is easy to see, if the total volume fraction of spheroids is specified, the minimization procedure (1) involves 2N − 1 fitting parameters. Generally, enlarging N for a given bandwidth, it becomes possible to obtain rms as small as desired. However, the fabrication of similar MMs becomes more and more challenging as N rises. Here, we take the moderate value of this parameter, N = 3, and the bandwidths of 10 and 15%.

A feature of any minimization procedure applied to the function η with the use of Eq. (6) is that several solutions for extracted geometrical parameters can occur. As an example, in Fig. 2 we show the results for ε_eff when Reε_eff has been fitted to zero with 10% bandwidth at λ₀ = 875 nm. For the first solution (solid line) rms = 0.043, while for the other (dashed curve) rms = 0.019. We also note that Imε_eff essentially depends on the choice of the solution; so, in our case it can differ from each other by as much as a factor of 2.5 within the actual band.

 

Fig. 2 The effective permittivity of a suspension of gold spheroids in a Teflon host with Reε_eff fitted to zero with 10% bandwidth at λ₀ = 875 nm. The parameters of the spheroids are L₁ = 0.0325, L₂ = 0.883, L₃ = 0.8945, f₁ = 0.077, f₂ = 0.0014, f₃ = 0.0016 (solid curves) and L₁ = 0.053, L₂ = 0.882, L₃ = 0.9907, f₁ = 0.0013, f₂ = 0.00068, f₃ = 0.078 (dashed curves) (here and below $L_i\equiv L_{\Vert }^{(i)}$).

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The different solution branches in the near-IR range are clearly seen in Fig. 3, where we show rms(λ₀) for the bandwidths Δ = 10% and 15%. In particular, two threefold branches occur in the range under consideration (for Δ = 15% we show only one curve for every branch). Moving to the visible range, rms (not shown here) rises rapidly. Two minima (at λ₀ ≈ 900 and λ₀ ≈ 1260 nm) are caused by the fact that the depolarization factor L⁽ⁱ⁾ of spheroids, comprising the dominant fraction in the suspension, takes its limiting value 1 (or 0) at these wavelengths after its monotonic increasing (decreasing). We note that the positions of these minima can be changed, if necessary, by replacing either inclusion or host material. For example, for copper in the near-IR range the real part of the permittivity is less negative than that for gold, and the above minima occur at longer wavelengths.

 

Fig. 3 The rms vs λ₀ for a suspension of gold spheroids in a Teflon host with Reε_eff fitted to zero with 10% and 15% bandwidths.

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As is well known, for many optical applications the dielectric losses play a critical role. To analyse the behavior of Imε_eff at different λ₀, in Fig. 4 we show the spectra of Imε_eff (λ) for Δ = 10% and several values of λ₀ up to those at which rms takes its minimum values. Only one of three solutions is shown for two solution branches (other solutions yield almost the same values of Imε_eff, as well as of Reε_eff). The results point to a decrease of dielectric losses as the central wavelength λ₀ rises.

 

Fig. 4 The imaginary part of the effective permittivity of a suspension of gold spheroids in a Teflon host with Reε_eff fitted to zero with 10% bandwidth. Solid curves and dashed curves correspond to two different branches. The respective values of λ₀ are shown in the inset.

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4. Discussion

Our further analysis involves dealing with strongly prolate and oblate spheroids. For convenience, let us denote strongly prolate spheroids (L_‖ ≈ 0) by ’p’, and strongly oblate spheroids (L_‖ ≈ 1) by ’o’. Close inspection shows that the curves 1, 2, and 3 in Fig. 3 correspond to (p-o-o), (p-p-o), and (p-p-p) arrays, respectively, while the curves 4, 5, and 6 correspond to (p-o-o), (p-p-o), and (o-o-o) arrays. Why such threefold branches occur? This issue looks interesting, and we address it in more detail.

For simplicity, we consider the problem in terms of real variables. Althougn we have performed our computations using complex values of the metal permittivity, the availability of the imaginary part (which is small as compared to the real part in this range) does not affect our reasoning significantly. Let us first take a look at the behavior of the Clausius-Mossotti factors for strongly prolate and strongly oblate spheroids. Because the permittivity of metal in the considered range is negative and large as compared to the permittivity of dielectric, it is easy to see that β_‖ is large and negative, while β_⊥ ≈ 2 when L_‖ is close to zero. This means that β_⊥ can be neglected when compared with β_‖ in this case. In contrast, when L_‖ is close to unity, β_⊥ is large and negative, while β_‖ ≈ 1. Thus, then β_‖ can be neglected when compared with β_⊥. Using this reasoning, one can see that the effective permittivity, represented by Eq. (6), for (o-o-o) arrays takes the form

In a similar fashion, Eq. (6) for (p-p-p) arrays takes the form

Here, we have taken N = 3 and have obtained two threefold branches. Similar reasoning shows that each branch becomes N-fold for an arbitrary number N. The above analysis shows that if the needed optical properties can be achieved for arrays of strongly oblate or strongly prolate spheroids, the same can be also achieved for mixtures of prolate and oblate spheroidal shapes. This leads to multiplicating solutions of the inverse problem.

It is interesting that Imε_eff correlates with rms, because Imε_eff within the actual band tends to decrease when ${\lambda }_0\to {\lambda }_0^{\mathit{min}}$ ( ${\lambda }_0^{\mathit{min}}\approx 900\hspace{0.17em}\text{nm}$ and 1260 nm for the two branches), see Figs. 2 and 3. Besides, we note that the function Imε_eff (λ) for various λ₀ is nonmonotonic within the actual band (see Figs. 2 and 4).

The presence of several solutions poses a question which one is most preferable. From a conceptual point of view, it makes sense to choose a solution for which rms is minimal. However, in practice this is not exactly so. The matter is that when ${\lambda }_0={\lambda }_0^{\mathit{min}}$, the depolarization factor L_‖ reaches its limiting value (0 or 1). This means that the basic geometry must include either infinitely long cylinders or thin sheets that seems to be impractical. Moreover, if L_‖ is too close to 0 (1), the spheroid aspect ratio becomes too large (small). Because the larger semiaxis of the spheroid must be much less than wavelength, this means that the spheroid (sharp needle or flat disk) becomes very thin. This, in turn, can present a considerable challenge to experimentalists. In addition, as one of the spheroid semiaxes is too small, due to electron scattering at boundaries, bulk value of the metal permittivity cannot be used when the electric field is applied along this axis. To take into account this effect, a more sophisticated homogenization technique must be developed.

By way of illustration, let us now consider specific examples. Assume that we are interested in designing ENZ MM with λ₀ = 875 nm and 10% bandwidth. Admitting that the maximum allowable size of the spheroids is 0.2λ, we evaluate it as 175 nm. Then, our calculations yield the following three solutions for the branch with predominating oblate spheroids: L₁ = 0.882, L₂ = 0.894, L₃ = 0.992, f₁ = 0.00067, f₂ = 0.00065, f₃ = 0.07868 for (o-o-o) array, L₁ = 0.053, L₂ = 0.882, L₃ = 0.9907, f₁ = 0.0013, f₂ = 0.00068, f₃ = 0.078 for (p-o-o) array, and L₁ = 0.053, L₂ = 0.0588, L₃ = 0.9896, f₁ = 0.0013, f₂ = 0.0014, f₃ = 0.077 for (p-p-o) array. For the branch with predominating prolate spheroids, we have: L₁ = 0.0336, L₂ = 0.053, L₃ = 0.0584, f₁ = 0.0736, f₂ = 0.00336, f₃ = 0.003 for (p-p-p) array, L₁ = 0.033, L₂ = 0.0584, L₃ = 0.8945, f₁ = 0.075, f₂ = 0.0029, f₃ = 0.0016 for (p-p-o) array, and L₁ = 0.0325, L₂ = 0.883, L₃ = 0.8945, f₁ = 0.077, f₂ = 0.0014, f₃ = 0.0016 for (p-o-o) array. Using the well known relationships between the depolarization factors of spheroids and their aspect ratios (see, e.g., [20]), we estimate the minimum size of the spheroids as 0.9 nm, 1 nm, and 1.2 nm in the former case and 25 nm, 13 nm, and 13 nm in the latter case, respectively. This obviously means that, due to existing manufacturing tolerances, composites with oblate spheroids predominating, although providing small rms and dielectric losses, are unfeasible for designing ENZ MMs with given λ₀ and bandwidth because they involve very thin spheroids (extremely flat disks). It should be noted that after decreasing λ₀, requirements for spheroid sizes become less stringent, while rms rises. So, at λ₀ = 750 nm, the solution for (o-o-o) array is L₁ = 0.827, L₂ = 0.845, L₃ = 0.932, f₁ = 0.0019, f₂ = 0.0018, f₃ = 0.076, while rms = 0.035, and the minimum size of the oblate spheroids is about 7 nm.

It is also of interest, from the point of view of realizability of designed composites, to assess the sensitivity of the obtained solutions to distribution of aspect ratios which is, obviously, always present in real composites. With this aim, we take the simplest (uniform) distribution of the form

As can be seen from Fig. 4, high losses can occur at least for one of the solution branches. It has been recently shown that ENZ hyperbolic MMs hold promise for realizing near-perfect absorption over a range of frequencies and subwavelength thicknesses [26]. In our case, high broadband absorption can be also achieved for subwavelength thicknesses (see Fig. 5).

 

Fig. 5 The intensity absorption for composite films of different thickness t made of gold spheroids in a Teflon host with Reε_eff fitted to zero with λ₀ = 900 nm and 10% bandwidth.

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Because here we do not tailor the magnetic response, impedance mismatching with free space causes the inevitable reflection at the vacuum-slab interface. This, in turn, limits the absorption by the slab. Losses reduce the transmission, but they also increase reflection (it is epecially high at long wavelengths, that results in an absorption decrease, as can be seen from Fig. 5). The high absorption within the band can be considered as a trade-off between the low reflection, which rises with losses, and low transmission, which drops with losses.

In some cases, losses have only a moderate impact on the performance of ENZ MMs, as it takes place, e.g., for tailoring the radiation phase pattern [27]. At the same time, it is well known that losses can substantially hinder the practical implementation of unique optical properties of many MMs and strongly limit the performance of corresponding devices. Dealing with metal constituents involves Ohmic losses related to undesirable conversion of electromagnetic energy into heat. And even nanostructures made of nearly lossless dielectrics suffer from transmission losses resulted from scattering and out-of-plane diffraction. Although overcoming the losses is, generally, a challenging task, there is a plenty of rooms to reduce or compensate them.

First of all, the negative impact of losses can be reduced by dealing with very thin films made of actual MMs (see, e.g., [28]). Another way is to use metals (or metal alloys) possessing lower intrinsic losses, e.g., silver instead of gold in the visible range. The use of superconducting materials with ultralow losses can be benefited at very low frequencies [29]. Furthermore, as has been recently suggested by Khurgin and Sun, in some synthetic structures the transition to the lossless mode can occur in the mid-IR range if the interatomic distances exceed certain values [30]. Finally, conceptually different solution could consist in using loss-compensating techniques involving dissipated structures combined with gain (optically active) media [31,32] or using parametric optical amplification [33].

Although the fabrication of MMs like those considered here is challenging, at present there exists a number of various techniques which allow to synthesize metal nanodisks [34, 35] and especially nanorods (see, e.g., [36,37] and references therein) with very reproducible sizes and shapes. Moreover, gold nanorods are commercially available with diameters from 10 to 50 nm, aspect ratios up to 20, and typical aspect ratio variation within 5% [38]. This allows us to hope that above MMs can be synthesized in the future.

5. Conclusion

In this paper we have considered the use of polydisperse suspensions of metal spheroidal particles for the purpose of designing broadband ENZ MMs. We introduced an efficiency of the permittivity nulling over a band (rms) and solved an inverse problem to extract the depolarization factors and partial volume fractions of the spheroids of different kind comprising designed MM. Considering gold nanospheroids in the Teflon host as an example, we have shown that the solutions of this problem form N-fold branches in the coordinates rms − λ₀, where λ₀ is defined as the centre of the actual band. For each branch, minima of the function rms(λ₀) can occur. Although the regime of operation close to the minima is best matched to the problem at hand, it is hardly feasible due to existing fabrication tolerances. In some cases, and especially for strongly oblate spheroids, aspect ratio distribution should be taken into account. In practice the parameters of suspensions must be chosen as a trade-off between small rms and dielectric losses, on the one hand, and practical realizability of designed MMs, on the other hand. As shown, the effect of high broadband absorption can be achieved with the use of subwavelength films made of designed ENZ MMs.