Making use of homogenization techniques, core-shell particle nanocomposites have been considered to estimate the efficiency of the effective permittivity nulling and enhanced absorption in terms of bandwidth. Core-shell metallodielectric spheres are shown to hold promise as building blocks for the design of metamaterials with the real part of the effective permittivity close to zero, as well as with a high absorption over a frequency band. In the former case, the imaginary part of the effective permittivity remains relatively high over the band, in particular, due to strong electron scattering at the shell boundaries. In the latter case, ultrabroadband metamaterial absorbtion can be realized which is resulted from individual absorption band overlapping.
© 2014 Optical Society of America
As nanotechnology becomes more and more mature, more and more challenging problems find their solutions. This, in turn, encourages us to pose problems which seemed desperately complex before. Recently, broadband epsilon-near-zero (ENZ) metamaterials (MMs) have received attention to improve the performance of regular (narrowband) ENZ MMs [1–7]. Although there should be many ways of designing such MMs, the choice of a specific design often appears to be ambiguous. In the one-dimensional (1d) and quasi-1d case, the problem admits an analytical solution and the bandwidth extension is relatively straightforward. In addition, accurate ab initio numerical simulation is available . The situation becomes more complicated in the 3d case, especially if we want to design an isotropic ENZ MM with the scalar effective permittivity. In this case, accurate numerical simulation is often prohibitively time and memory consuming, while analytical solutions are available for certain geometries only and may need additional justification and an analysis of their applicability [4, 7]. The first goal of this work is to consider an example of such geometry which allows one to directly evaluate the effective permittivity and to design broadband ENZ MMs. Furthermore, the design of ENZ MMs can be closely connected to the design of highly absorbing MMs [2, 7–11]. However, in the 3d case this issue has not been properly addressed so far. Our second goal is to partially fill this gap.
In the present study we consider a generalized version of the Maxwell Garnett theory (MGT) of homogenization. Originally, this theory was developed for a dilute suspension of small homogeneous spheres embedded in a host medium . Despite its completely phenomenological character, as was noted , “Suprisingly enough, the Maxwell Garnett theory works very well even if the hypotheses that 2πR/λ ≪ 1 and f ≪ 1, where f is the volume fraction occupied by the spheres, are no longer valid. The Maxwell Garnett theory has been shown to describe the effective medium properties of cubic photonic crystals within a few percent of exact result till the first stop gap, and that even in the close-packed case. The latter also holds for core shell spherical particles”. In contrast to the recent study , where the design of the broadband ENZ MMs has been considered with the use of multiphase arrays of homogeneous (metal alloy) spheres, here we deal with multiphase arrays of shelled spheres.
Whites, using a T-matrix solution for lattices of dielectric and conducting spheres, has shown excellent agreement between the MGT and his results up to as large inclusion volume fraction as 0.45 . For random sphere arrays, the MGT descibes the effective permittivity very well as the dielectric contrast is not too high and the spheres are well separated . For superconducting inclusions (the infinite dielectric contrast), the MGT withstands the most stringent test for the volume fraction up to 0.3 when compared with the strong-contrast expansion in terms of three-point correlation function . At the same time, for disordered systems, a red shift and broadening of absorption peaks occur as compared to the MGT; these effects come into particular prominence as filling factor reaches 10 – 20%. At the higher filling factors, multipolar effects come into play and need to be taken into account .
Our homogenization techniques deals with metal-covered dielectric spheres. Moreover, the spheres are of different kinds. This is necessary in order to provide a sufficient number of degrees of freedom that, in turn, makes it possible to properly adjust the effective permittivity within a frequency band. Besides, it is imperative that dealing with metal phase demands the need to take into account material loss. Although the realization of all-dielectric (lossless) zero-index MMs has been recently reported, they are narrowband rather than broadband . At the same time, due to recent interest in lossy ENZ media (see, e.g., [8–10,19,20]), in this paper the effect of material loss has been considered in more detail. Although 3d isotropic ENZ MMs can in themselves reveal high broadband optical absorption , their geometry can be further optimized to improve absorption efficiency. So, dealing with the same basic geometry, we consider the problem of designing broadband highly absorbing MMs, which are especially important for the development of various solar thermovoltaic and solar-thermal energy convertors (see, e.g., [21–23] and references therein).
2. Basic technique and approximations used
In our consideration, the problem of broadband permittivity nulling over the frequency band [ω1, ω2] can be formulated as finding a minimum of the objective function
Alternatively, the problem of broadband absorption enhancement is simply to minimize the function7]. Thus, within the framework of our approach, both problems reduce to the proper choice of the composite constituents and its geometry.
In the ideal case, geometry under consideration may be imagined as a “supercrystal”. It deals with particles (spheres) of different kind which are embedded in a dielectric host (matrix) (see Fig. 1). If so, a representative volume element of such a polydisperse supercrystal may be introduced which is characterized by a scale Λ such that Λ ≪ λ, but the volume Ve = Λ × Λ × Λ still contains a lot of the spheres which form a cubic lattice. The volume fraction (filling factor) of the spheres of certain kind within each representative volume element is fixed. It is necessary to keep in mind that the above supercrystalline structure can be difficult to fabricate even it is feasible. It may be assumed, however, that some disordering should not significantly affect the validity of our model, if only the particles are still well-separated. Thus, the supercrystal is considered to consist of the representative volume elements such that for each of them ∫Ve ε(r)E(r)dr/∫Ve E(r)dr = εeff, where ε(r) and E(r) are the local permittivity and electric field, respectively, and εeff is the effective permittivity.
It should be noted that there is at present no rigorous theory of homogenization for such supercrystals. Because of this, in our further analysis we use Maxwell-Garnett type approximations. As was noted above, the standard MGT is believed to be correct for dilute suspensions only, when the total filling factor f 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). This follows from the fact that when f is small, the real part of the effective permittivity becomes positive everywhere within the band, and εeff approaches the host permittivity as f → 0. Although here we restrict our attention to the long-wavelength limit (the quasi-static approximation), concurrently the MGT can be generalized to take into account the multipolar interaction between the particles which manifests itself at higher particle concentrations.
Metal-coated spherical particles (plasmonic nanoshells) are of interest because their optical properties are highly sensitive to the inner and outer dimensions of the shell layer. Because their plasmon resonance can be tuned in wide limits, the nanoshell-based composites hold promise for various applications requiring broad bandwidth . In what follows we consider rms as a function of the central wavelength λ0 = 2πc/ω0 for different values of the dimensionless bandwidth.
One of approaches to calculate the effective permittivity of arrays of shelled spheres has been proposed by Gao et al. . However, it is limited to the case of a very thin and highly conducting interface. For confocal shelled ellipsoids, a self-consistent approach has been developed by Tinga et al. . Using an analytical approach by Harfield , Bowler has obtained an expression for the effective permittivity of shelled spheres arranged on a simple cubic lattice within the framework of the interacting dipole approximation .
In our opinion, the consideration of this problem is more convenient in terms of the so-called “equivalent dielectric constant” concept suggested by Bilboul . The equivalent dielectric constant (permittivity) has been introduced for a uniform ellipsoid which is indistinguishable from the original confocal shelled ellipsoid (more exactly, the value of the equivalent permittivity “is such that if the permittivity of the surrounding medium is equal to it, the field and the potential at any point in the medium is unperturbed by the introduction of the double-layer ellipsoid” ). Solving Laplace’s equation in ellipsoidal coordinates, it may be shown that the induced electrostatic potential outside the small shelled particle is indistinguishable from that of the equivalent homogeneous particle. The same result in the particular case of a double-layer sphere was derived by James Clerk Maxwell in XIX century who considered the problem in terms of the equivalent conductivity . It should be noted that the validity of replacing the shelled sphere by a homogeneous sphere with the equivalent dielectric constant was tested using ab initio FDTD simulation, and a reasonable accuracy within a few percent was obtained that is comparable with the accuracy of this simulation technique . The above replacement, in turn, allows one to introduce an equivalent polarizability of the inclusions and to generalize the original MGT. For the spheres of N kinds, the effective permittivity can be evaluated as [4, 32]Eq. (3) coincides with Eq. (16) given in , as well as with Eq. (4) given in .
For higher particle concentrations, more sophisticated approaches are necessary to accurately describe the effective permittivity. In particular, for cubic lattices of spheres we would like to note the results by McKenzie and McPhedran [33, 34], who addressed the problem in terms of the electric conductivity, as well as by Sangani and Acrivos , and Cheng and Torquato , who addressed the problem in terms of the thermal conductivity. Finally, the effective permittivity for arrays of shelled spheroids arranged on a simple cubic lattice was obtained by Harfield who treated the shelled particles as interacting multipole sources .
Due to the symmetry of cubic arrays, multipole of the lowest order (after the dipole) is the octupole. Taking into account octupole correction only, the effective permittivity can be written down as 37], the effective permittivity sufficiently depends on the specific form of the two-particle distribution function for the volume fractions larger than 0.2. This means that because some degree of disorder is always present in real arrays, caution must be exercised in using Eqs. (3) and (5) for such high concentrations.
For very small spheres, and especially for very thin shells, electron-interface scattering at the inner and outer interfaces of the metal shell can become significant that results in the reduced mean free path of electrons. In this case, thin shell corrections to the bulk permittivity of the metal are necessary. To accomodate the reduced mean free path due to the boundaries of the thin metal shell, a modified collision frequency γ can be introduced as
It should be noted that finding the reduced effective mean free path for the shelled sphere geometry is not trivial. In detail, this issue has been addressed by Moroz , who provided strong arguments in favor of the so-called billiard model for Leff . According to this model, the effective mean free path can be written as(7) must be taken into account if either r1 is too small or q → 1.
3. Numerical results
In what follows, we take gold and silver as the metal constituents. Their permittivities have been taken using the Drude-critical points model which is shown to be rather accurate at wavelengths above 400 nm . Teflon (ε0 = 1.823), silica (ε0 = 2.25), and a hypothetical dielectric with ε0 = 2.5 have been considered as host media. Silica has been taken as the core phase. We note that both silica-gold and silica-silver shelled nanospheres have already been synthesized in many laboratories (see, e.g., [41, 42]).
Each sphere can be characterized by two geometrical parameters (radii of the inner and outer surfaces of the sphere). Dealing with the spheres of different kind allows one to extend the bandwidth, because this enlarges the number of fitting parameters entering the objective functions (1) and (2), but this also makes further analysis more complicated. In addition, the more different shelled spheres are present in the array, the more complicated fabrication technique is. Due to that, in what follows we limit ourselves to N = 3. Besides, to simplify the further analysis, we fix r1 = 0.04λ0, considering qi and fi as fitting parameters. To start, it looks constructive to see first what happens in the case of monodisperse assembly (N = 1).
3.1. Preliminary analysis: Monodisperse arrays
The use of monodisperse core-shell nanosphere arrays to implement ENZ MMs have been recently discussed by Monti et al., who have shown that the imaginary part of the effective permittivity can be moderate for metal-dielectric core-shell MMs in the visible range in the ENZ regime . In particular, the authors have demostrated this for highly concentrated arrays (the filling factor of 0.3) of silica particles covered with silver. Although their results pertain to the narrow-band case only (Δ = 0), when the real part of the effective permittivity is nulled at one frequency, and no scattering at shell boundaries has been taken into account, these results can be considered as the starting point for our subsequent study. So, for better understanding of behavior of the effective permittivity, we show it for the case of arrays of silica-silver core-shell spheres of one kind for different values of the core volume fraction q (see Fig. 2). As one can see, Reεeff nulls at two wavelengths at each q. If they are, say, λ01 and λ02, Reεeff (λ01)=Reεeff (λ02) = 0, and λ01 < λ02, then Imεeff (λ01) <<Imεeff (λ02). To obtain ENZ MM, of interest to us is the crossover wavelength λcros ≡ λ01, at which Imεeff is not so large as at λ02. At the same time, for the highly absorbing MM, of primary interest is the wavelength λ02, at which the dipolar resonance of shelled spheres occurs. Our basic idea can be described in general form as follows. Broadband ENZ and high absorption regimes can be realized for polydisperse core-shell nanoshere arrays, if a few closely-spaced crossover or resonance wavelengths, respectively, lie within the band of interest.
Taking into account the importance of material losses for ENZ MMs, in Fig. 3 we show the function of Imεeff (λcros) in the visible and near-IR range for different host media with f = 0.15 (solid curves) and f = 0.3 (dashed curves). We note the rather high value of Imεeff (λcros) which, even reduced after enlarging the sphere volume fraction f up to 0.3, exceeds 0.5 for all the above hosts.
3.2. ENZ metamaterials
The effective permittivity of arrays of silica-silver core-shell spheres embedded in a host with ε0 = 2.5 when Reεeff has been fitted to zero with 10% and 15% bandwidths at λ0 = 600 nm is shown in Fig. 4. The root mean squares are rms = 0.016 (Δ = 10%) and rms = 0.065 (Δ = 15%). It is interesting that while the parameters qi (which specify the particular sphere polarizabilities αi) are almost the same for both bandwidths, the parameters fi (which specify “the weight” of the spheres of certain kind) are sufficiently different.
When the total filling factor f grows further, our calculations (not shown here) testify that rms first drops and then increases. For example, for simple cubic lattice at Δ = 10% and λ0 = 600 nm, rms = 0.013 at f = 0.2, 0.005 at f = 0.28, 0.009 at f = 0.35. The losses (Imεeff) tend to lower within the actual band, especially at low frequencies, but they still remain rather high.
To assess the relationship between the bandwidth and losses, in Fig. 5 we show the spectra of Imεeff for the bandwidths Δ = 10%, 5%, and 2.5% at λ0 = 600 nm and f = 0.15. These results evidence that extending bandwidth results in a simultaneous increase in dielectric loss within the actual band.
3.3. Highly absorbing metamaterials
As all above MMs reveal high losses, it seems constructive to consider their potential for the design of highly absorbing broadband MMs. With this aim, we minimize the objective function (2) to maximize absorption and find geometrical parameters of the quasicrystal. As an example, in Fig. 6 we show the absorption spectra for slabs with the thickness d = λ0 = 600 nm made of core-shell silica-silver spheres embedded in different hosts. As one can see, Imεeff exceeds 0.95 within the band. In all cases, rms ≃ 0.045.
It is of interest to see what happens if we further extend the bandwidth and if we change silver with gold. So, in Fig. 7 we show the absorption spectra for the MM slabs of thickness d = λ0 with Δ = 50% at f = 0.2 and ε0 = 1.823 for silver (rms = 0.065) and gold (rms = 0.07) nanoshells. Comparing the results for the silver nanoshells, it comes as a surprise that so significant bandwidth extension causes so slight deterioration of rms. If f is increased, rms can be further improved. For example, at f = 0.4 (not shown here), rms = 0.06 for silver nanoshells and rms = 0.063 for gold nanoshells in the case of simple cubic lattice.
First of all, as it follows from Figs. 2 and 3, even for monodisperse arrays (the narrowband case) the designed ENZ MMs are characterized by rather high ε″eff(λcros)=Imεeff (λcros), i.e., by high material losses. For polydisperse arrays (the broadband case) the losses rise still further (see Fig. 5). Obviously, this results from the following two reasons. First, the ENZ band appears close to individual resonances of core-shell spheres (wavelengths λ02), where ε″eff=Im(εeff) is high. This is in contrast to the quasi-1d case, where the ENZ regime realizes out of resonances . Second, strong electron scattering at shell boundaries corrects for the imaginary part of the metal permittivity according to Eq. (7) as the effective mean free path Leff becomes less than the electron mean free path for corresponding bulk materials (for example, the latter is about 40 nm for gold and about 50 nm for silver). This, in turn, increases the material losses of the designed MMs. At the same time, in the quasi-1d case the interfacial scattering is low, as the electric field is directed along phase boundaries .
For simplicity and convenience, as this analysis can be done analytically, consider the behavior of dielectric loss at the crossover wavelength in the narrowband case. As it follows from Eq. (3), if Imεeff = ε″eff ≪ Reεeff = ε′eff, ε′eff (λcros) = 0 when α′(λcros)=Reα(λcros) ≃ −1/2 f, orEqs. (10) and (11) may be rewritten as Eq. (13) into Eq. (12), one has Eq. (15) in the form of
The total volume fraction of the spheres, as well as constituent permittivities are the main parameters whose influence on the performance of designed MMs deserves attention. Our basic geometry, the array of shelled spheres, involves 3 phases: host, shell, and core media. This means that, generally, we could examine the influence of three material paramaters (permittivities) on the performance. Consider the influence of these parameters in their order.
- (1) As one can see from Fig. 3, in the narrowband case, an increase of the total volume fraction of the shelled spheres, f, leads to a decrease in ε″eff at the crossover wavelength λcros. This agrees with earlier results by Monti et al. . As we made sure, this also takes place in the broadband case. At first sight, this looks unnatural, because, as the content of the absorbing phase f1 = f (1 − q) grows, the losses should also grow. However, as one can see from Eq. (14), this is not so. In fact, the dependence of ε″eff (λcros) on f is governed by the product of two factors, (1 + f + 1/4f)ε ″e (λcros). When f grows (but remains not too large), the first factor drops. As to the second factor, it also drops due to an increase in q (this accompanies the increase in f). When f → 0, λcros disappears and Eq. (14) becomes inapplicable.
In the broadband case, the situation becomes more complicated. From Eq. (3), one hasEq. (17) reveals that ε″eff is roughly proportional to ∑ ifiα″i. The partial polarizabilities αi exhibit resonance behavior and can vary in wide limits within the band; however, their imaginary parts are relatively large near the resonances only. For monodisperse arrays, the resonances occur near λ = λ02, where ε″eff is high, as can be seen from Fig. 2. Out of the resonance (at λ = λcros), ε″eff is moderate. At the same time, for polydisperse arrays, we deal with the sum of partial polarizabilities, which does not exhibit resonance behahior and varies only slightly within the band. The imaginary part of this sum is much larger than that of any partial polarizability. Thus, broadening bandwidth, which involves several plasmon resonances within the band, is favourable for increasing losses. For illustration, in Fig. 8, we plot fiα″i and their sum for the same shelled spheres as in Fig. 4. We note that the contour of fiα″i in Fig. 8 reproduces with good accuracy that of ε″eff shown in Fig. 4.
- (2) The permittivity of the host medium. As can be seen from Eq. (9), the effect of this permittivity should be opposite to the effect of the increase of f. This means that as ε0 grows, q drops off, and ε″eff also drops. However, for large q the condition ε″eff ≪ ε′eff is no longer valid, and hence Eq. (9) becomes inapplicable. As a result, ε″eff can grow with ε0, as can be seen from Fig. 3.
- (3) The permittivity of the shell medium. As can be seen from Eq. (15), the effect of the shell (metal) choice on ε″eff(λcros) depends on the interplay between ε″1 and ε′1. Our calculations (not shown here) evidence that if we change the silver shell with a gold shell, this results in a strong increase of ε″eff(λcros) at λ0 < 600 nm; this happens due to strong interband transitions which occur in bulk gold at these wavelengths. At high wavelengths, however, ε″eff(λcros) for the gold shell is slightly smaller than that for the silver shell.
- (4) The permittivity of the core medium. Our calculations (not shown here) evidence that as ε2 grows, ε″eff (λcros) grows, too. This happens due to an increase of q in Eq. (16).
The above discussion shows that, to minimize ε″eff(λcros), the permittivities ε0 and ε2 should be as small as possible, while the volume fraction of the spheres, f, should be large, but not too large, to avoid multipolar effects. At the same time, the sphere radius, r1, should be large, to reduce interfacial scattering, but not too large, to remain within the scope of the quasistatic approximation. Of course, it is necessary to keep in mind that dealing with regular metallodielectric MMs does not allow to make dielectric losses as small as desired, because of the existing lower bound on ε″eff [1, 6].
Although the use of core-shell sphere arrays does not allow to design broadband ENZ MMs with low losses, this issue is not critical in some cases . Moreover, as was noted earlier, losses can be desirable, especially for light absorption and trapping [8–10].
The proposed technique of designing highly absorbing broadband MMs involves the use of strong electromagnetic resonances. Usually, the bandwidth of such absorbers is relatively narrow (typically, within 20% of the central frequency ). To extend the bandwidth, a few resonances can be combined when the representative volume element includes shelled spheres of different kind. If the resonances are located closely enough, they can overlap providing a broad bandwidth. In Fig. 7 we demonstrate the absorption spectra for silver and gold nanoshells with 50% bandwidth when the representative volume element includes the shelled spheres of three kinds. The absorptance within the band exceeds 90%. It is clear that further extending band-width comes at the expense of reduced absorptivity. At the same time, we note that lowering ε0 improves the absorption efficiency (extends the bandwidth), as can be seen from Fig. 6.
The use of metal nanoparticles for the design of broadband absorbers has been shown to be promising earlier (see, e.g., ). However, dealing with nanoparticles of one kind still limits the potential of such an approach. Finally, it should be noted that for both monodisperse and polydisperse sphere arrays, the absorption coefficient cannot be too close to unity. For example, the absorptance shown in Fig. 6, does not exceed 0.96 within the absorption band. To further improve absorptivity, the impedance of an absorber must be matched to that of free space . Otherwise, a parasitic reflection within the absorption band is inevitable. At the same time, there exists a conceptually different mechanism of a high absorption, involving the excitation of the Berreman mode and ENZ mode [10, 11]. Although the authors of the above papers have considered the narrowband case only, we believe that such an approach can be extended to realize highly absorbing broadband MMs.
Our approach is valid in the quasistatic approximation only; this imposes constraints on the maximal size of the spheres. One can expect, however, that increasing sphere size beyond the limit r1 ≪ λ would allow to further improve the broadband absorption efficiency, in particular, due to trapped light mode . One more limitation is that it does not take into account possible effects of nonlocality. An accurate description of these effects involves the use of first-principles methods and goes beyound the scope of our paper. Nonlocal effects, however, are considered to be sufficient for metallic shells of ultrasmall dimensions only, typically, less than 10 nm (see, e.g., ).
In this paper, we have considered the potential of polydisperse shelled sphere arrays in the quasistatic regime for their use as broadband ENZ and highly absorbing MMs. The our main findings can be summarized as follows. The real part of the effective permittivity can be successfully nulled over a broad frequency band; the imaginary part, however, remains relatively high within the band. At the same time, ultrabroadband absorption can be realized for such arrays as a result of individual absorption band overlapping.
A.P. acknowledges partial support from Biofrontiers Institute, UCCS and from CRDF ( UKC2-7071-CH-12), and Y.C.C. acknowledges support from National Science Council of the Republic of China under Contract No. NSC 101-2112-M-001-024-MY3.
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