Metallo-dielectric core-shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials

The open–access journal for physics New Journal of Physics Metallo-dielectric core–shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials R Paniagua-Dom´ ınguez1,FL´ opez-Tejeira1, R Marqu´ es2 and J A S´ anchez-Gil1,3 1Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cient´ ıficas, Serrano 121, 28006 Madrid, Spain 2Departamento de Electr´ onica y Electromagnetismo, Universidad de Sevilla, 41012 Sevilla, Spain E-mail: [email protected] New Journal of Physics 13 (2011) 123017 (15pp) Received 18 August 2011 Published 9 December 2011 Online at http://www.njp.org/ doi:10.1088/1367-2630/13/12/123017 Abstract. Materials showing electromagnetic properties that are not attainable in naturally occurring media, so-called metamaterials, have been lately, and still are, among the most active topics in optical and materials physics and engineering. Among these properties, one of the most attractive ones is the subdiffraction resolving capability predicted for media having an index of refraction of −1. Here, we propose a fully three-dimensional, isotropic metamaterial with strong electric and magnetic responses in the optical regime, based on spherical metallo-dielectric core–shell nanospheres. The magnetic response stems from the lowest, magnetic-dipole resonance of the dielectric shell with a high refractive index, and can be tuned to coincide with the plasmon resonance of the metal core, responsible for the electric response. Since the response does not originate from coupling between structures, no particular periodic arrangement needs to be imposed. Moreover, due to the geometry of the constituents, the metamaterial is intrinsically isotropic and polarization independent. It could be realized with current fabrication techniques with materials such as silver (core) and silicon or germanium (shell). For these particular realistic designs, the metamaterials present a negative index in the range of 1.2–1.55 µm. 3Author to whom any correspondence should be addressed. New Journal of Physics 13 (2011) 123017 1367-2630/11/123017+15$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 Contents 1. Introduction 2 2. Optical properties of metallo-dielectric core–shell meta-atoms 3 2.1. Magnetic resonance of a high-permittivity shell ................. 4 2.2. Electric and magnetic resonances on core–shell systems ............. 5 3. Calculation of effective parameters 7 3.1. Effective parameters of a random arrangement of core–shell nanospheres . . . 8 3.2. Effective parameters of a cubic periodic arrangement of core–shell nanospheres 8 4. Concluding remarks and fabrication possibilities 11 5. Methods 12 5.1. Effective index retrieval procedure ........................ 12 Acknowledgments 14 References 14 1. Introduction Once the possibility of building a negative-index metamaterial (NIM) was proven in the microwave regime [1], extraordinary effort was made to obtain analogous behaviour for increasingly higher frequencies up to the visible range of the electromagnetic spectrum [2,3]. The major obstacle found when trying to translate ideas to higher frequencies was how to achieve a strong diamagnetic response in the systems designed. Many attempts to tackle the problem were simple miniaturizations of the canonical designs employed in those first metamaterials operating in microwaves. As an example, the split-ring configurations, or slight variations, were reduced up to the nanoscale to obtain such a magnetic response. Apart from the fundamental limitations inherited from those designs, e.g. anisotropy, and the increasing complexity of the fabrication procedures, several drawbacks have been found in this process, some of them being a consequence of the different behavior of metals when excited with higher frequency waves [4]. Precisely, this different behavior also inspired some authors to search for different configurations intended to exploit the plasmonic response to obtain artificial magnetism. In many of them, artificial magnetism is due to coupling between different plasmonic structures, their drawback thus being the high losses within metallic parts [5]. In most cases, the proposed designs are restricted to operate under certain polarization and incidence conditions [6–8] or are not truly three-dimensional (3D) materials [9,10]. Moreover, in many cases, the behavior of the proposed designs stems from coupling between the different constituents, thus making particular arrangements necessary. As a consequence, spatial dispersion effects often appear due to the propagation of waves in the lattice. Lately, some approaches based on plasmonic waveguides supporting negative-index modes have pushed the frequencies in which left-handed (LH) behavior is obtained well within the visible spectrum. However, propagation inside such materials would be limited to the propagation length of the plasmon in the waveguide, thus making the design a single-layer device, lacking true threedimensionality [9,10]. In addition, various attempts have been made towards NIM by exploiting magnetic resonances occurring in structures made of high-permittivity materials [11–13]. Some of them combine these structures with secondary structures providing the electrical response [14–17] or embed them in a metallic host medium [18], thus having inherent high New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 3 losses. Here we report a design that tackles many of the previously mentioned limitations, namely, a totally 3D isotropic NIM, operating at optical frequencies, whose response is due to every isolated ‘meta-atom’. Therefore, no particular arrangement of them is needed. We study the possibility of using a spherical core–shell configuration to obtain with a single structure both electric and magnetic responses. The core, being metallic, is responsible for the electrical response, while the shell, made of a high-permittivity dielectric, provides the strong diamagnetic response. An extension of Mie theory is exploited to rigorously determine the scattering properties and resonances of the whole spherical core–shell configuration, which essentially determines the effective material properties [19]. Calculations for core–shell structures built up with realistic materials (Ag@Si and Ag@Ge) demonstrate the possibility of obtaining NIM operating at 1.2µm–1.55 µm. Since both responses are attained directly from a single constituent, no particular arrangement of the inclusions is needed. In order to extract the effective parameters of the metamaterial, we will assume both a random and a simple cubic distribution. In the case of a random distribution, the Lorentz–Lorenz theory is applied, leading to simultaneously negative effective permeability and permittivity for several filling fractions. In the case of a simple cubic lattice arrangement, the finite-element method is applied to carry out numerical simulations, which fully account for interaction between the periodically arranged constituents. Effective material constants are then extracted through the standard S-parameter [20,21] retrieval procedure, and are tested to fulfill causality and passivity, thus confirming a true double-negative index. Due to the spherical symmetry of the constituents, the metamaterial response will be essentially isotropic and polarization independent. 2. Optical properties of metallo-dielectric core–shell meta-atoms Let us examine the scattering of a plane electromagnetic wave (wavelength λ) from a spherical core–shell particle without any approximation, which indeed can be done analytically as an extension of Mie theory, obtained first by Aden and Kerker [22]. Figure 1(a) depicts the geometry of one of the basic constituents or ‘meta-atoms’: a high-permittivity () dielectric shell is considered with outer radius Rλand thickness T,cand 0being the dielectric constants of the core and the surrounding medium, respectively. The scattering and extinction efficiencies can be expressed in terms of the material and geometrical parameters through the scattering coefficients aland bl(which represent, respectively, the different electric and magnetic multipolar contributions) as Qsca =2 y2 ∞ X l=1 (2l+ 1)(|al|2+|bl|2), (1) Qext =2 y2 ∞ X l=1 (2l+ 1)<(al+bl), (2) where y=k R. The mentioned scattering coefficients can be written in terms of the spherical Bessel functions of the first ( jl(x)) and the second (yl(x)) class and depend on c/0,/0,Rin and R.<denotes the real part. Their explicit form is al=ψl(y)[ψ0 l(ny)−Alχ0 l(ny)]−nψ0 l(y)[ψl(ny)−Alχl(ny)] ξl(y)[ψ0 l(ny)−Alχ0 l(ny)]−nξ0 l(y)[ψl(ny)−Alχl(ny)],(3) New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 4 (a) (b) (c) (d) (e) Figure 1. (a) The geometry of the problem. (b) Dipolar magnetic contribution to the total scattering efficiency of a silicon (Si) nanoshell with outer radius R=170 nm as a function of the inner radius (Rin) and wavelength of the incident light. (c) Total scattering efficiency, together with the dipolar electric (a1) and magnetic (b1) contributions to the scattering efficiency for the inner radius Rin =45 nm (indicated in (b) by a dashed white line). (d, e) Near-field plots at the magnetic resonance for the Si shell of (c). (d) Norm of electric field. (e) Out- of-plane component (only nonzero component of the incident magnetic field) of H, together with the electric displacement field in white arrows. bl=nψl(y)[ψ0 l(ny)−Blχ0 l(ny)]−ψ0 l(y)[ψl(ny)−Blχl(ny)] nξl(y)[ψ0 l(ny)−Blχ0 l(ny)]−ξ0 l(y)[ψl(ny)−Blχl(ny)].(4) The Ricatti–Bessel functions introduced are ψl(z)=zjl(z),χl(z)=−zyl(z)and ξl(z)= zh(1) l(z), where h(1) l(z)=jl(z)+ iyl(z)is the spherical Hankel function of the first class. The coefficients Aland Blare Al=nψl(nx)ψ0 l(ncx)−mcψ0 l(ny)ψl(ncx) nχl(nx)ψ0 l(nx)−ncχ0 l(ny)ψl(ncx),(5) Bl=nψl(ncx)ψ0 l(nx)−ncψl(ny)ψ0 l(ncx) nχ0 l(nx)ψl(ncx)−ncψ0 l(ncy)χl(nx),(6) where x=k Rin,n2=/0and n2 c=c/0. It can immediately be realized that all information about material and geometrical properties of the core is contained in these Aland Blcoefficients. 2.1. Magnetic resonance of a high-permittivity shell To better understand the underlying physics, we first characterize the magnetic resonance of a nanoshell for a real dielectric material with a high refractive index. We choose silicon (Si) the refractive index of which can be considered constant, n=√/0∼3.5, and lossless, within the near-infrared (IR) range λ=1–2 µm. In figure 1(b), the contribution is plotted to the total scattering efficiency of the magnetic dipolar term (b1) as a function of Rin and the incidence wavelength. It is the dominant contribution, as can be seen in figure 1(c) for the specific case of Rin =45 nm. A resonance can be clearly seen, the wavelength of which corresponds to that New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 5 of the Si compact sphere [23] when Rin →0 and starts to redshift when the shell thickness is thin enough ( T.R/3) and decreases. Thus the magnetic resonance of the Si nanoshell can be tuned within a certain range of wavelengths larger than that of the Si sphere resonance. The behavior of the electromagnetic (EM) fields at the magnetic resonance has been calculated by full-wave (finite-element method) numerical simulations [24]. The result is depicted in figures 1(d) and (e). It can be observed that the electric displacement field rotates in planes parallel to the equator, thus inducing a strong magnetic moment. This pattern clearly shows that the electrical displacement current −iωεEis strongly confined inside the shell, and rotates along the φdirection around the incident H-field. Such behavior reveals that the resonance is qualitatively quite similar to the first resonance of a dielectric ring, already reported [25], following an LC model with an inductance L∝µ0Rassociated with the circulation of the displacement current, and a capacitance C∝εRassociated with the electric energy confined in the shell. At resonance, a strong magnetic moment along the direction of the incident H-field is generated in the same qualitative way as in the dielectric ring analyzed [25]. 2.2. Electric and magnetic resonances on core–shell systems Next we analyze the mutual influence of a metallic core and a dielectric shell. With regard to the plasmon resonance of the metallic core (i.e. collective oscillations of conduction electrons), it is well known that for small spheres (Rin λ) of dielectric permittivity c(ω), embedded in a medium with dielectric constant 0, the induced dipole moment is resonant at the frequency ωLSPR such that c(ωLSPR)=−20[26]. Therefore, when the metallic sphere is coated with a thick dielectric layer of permittivity , the quantity driving the resonance condition is not c itself but, instead, the ratio between the permittivities of the core and the coating. That is, for a small metal core, the resonance occurs when c/ ∼−2 is fulfilled, thus redshifting the localized surface-plasmon resonance (LSPR) with respect to that in vacuum. The question now arises as to whether or not the shell magnetic resonance is preserved when the core is metallic. Since the confinement and rotation of the electric field inside the coating is directly related to the jump conditions for the normal component of the field between the shell and the surrounding medium [25], and since the metal is expected to avoid penetration of the field inside, no significant changes of the magnetic resonance behaviour are expected, at least for relatively small metallic cores. This will be revealed below, where we plot the scattering efficiency together with the electric and magnetic dipolar contributions. Such behaviour makes it possible to still predict the appearance of the magnetic resonance even if a metallic core is present. Furthermore, the electric and magnetic responses of a metal–dielectric core–shell nanosphere can be tuned to make both resonances coincide. Let us take silver (Ag) as the material to build up the core and Si again as a high-permittivity dielectric for the shell. Complex Ag permittivity values are taken from tabulated data [27]. Our goal is to obtain simultaneously both the resonances in the near-IR range of the electromagnetic spectrum. The LSPR will be redshifted to that wavelength such that Ag r∼−2Si. This happens at a wavelength of about 720 nm. Nevertheless, we have to keep in mind that, strictly, this would only be a good approximation for very small metallic particles with relatively thick coatings. Since we want to push the electric resonance to the IR to make it coincide with a magnetic one stemming from the Si shell, we can take increasingly larger radius for the core. When the size of a metallic particle is increased, there is a redshift in the resonance wavelength that can be explained in terms of depolarization effects [26]. New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 6 Figure 2. Optical properties of Ag@Si and Ag@Ge core–shell nanospheres. (a–c) Dipolar electric contribution (a), dipolar magnetic contribution (b) and total scattering efficiency (c) of an Ag@Si core–shell nanosphere with outer radius R=170 nm as a function of the inner radius (Rin) and wavelength of the incident light. (d) Extinction and scattering efficiencies, together with the dipolar electric (a1) and magnetic (b1) contributions to the scattering efficiency for inner radius Rin =47 nm (indicated in (c) by a dashed white line). (e–g) Dipolar electric contribution (e), dipolar magnetic contribution (f) and total scattering efficiency (g) of an Ag@Ge core–shell nanosphere with outer radius R=190 nm as a function of the inner radius (Rin) and wavelength of the incident light. (h) Extinction and scattering efficiencies, together with the dipolar electric (a1) and magnetic (b1) contributions to the scattering efficiency for the inner radius Rin =55 nm (indicated in (g) by a dashed white line). Figures 2(a)–(c) depict both, the electric dipolar and the magnetic dipolar contributions to the scattering efficiency, together with the total scattering efficiency for an Ag@Si core–shell system of outer radius R=170 nm as a function of Rin and the incidence wavelength. One can clearly see an overlap between the electric and magnetic resonances. It occurs within 1150 and 1300 nm and for inner radius Rin =40–50 nm. We have explicitly plotted the case Rin =47 nm (figure 2(d)). An interesting feature that can be observed is that the magnetic resonance disappears as the inner radius increases. It can be explained by the fact that the electric displacement field in the core rotates in the opposite direction to that in the shell, thus reducing the total magnetic moment generated. Another effect can be seen in the electric dipole contribution. As the inner radius increases the resonance broadens and redshifts as expected. Interestingly, this behavior changes when the thickness is comparable with Rin, and the resonance starts to blueshift. This effect can be attributed to the fact that, as the dielectric shell becomes thinner, the resonance wavelength tends to that of a sphere without coating, thus blueshifting. Finally, it is important to note that the specific wavelengths at which the electric New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 7 (a) (b) Figure 3. Near-field plots for an Ag@Si shell with R=170 nm and T=123 nm. (a) Norm of the electric field in log10 scale at combined electromagnetic resonance and when the system is excited by a purely electric excitation (topright, E) and by a purely magnetic one (bottom-right, H). Color scale, incidence and polarization are preserved in all figures. (b) Out-of-plane component of H, together with the electric displacement (D) field in black arrows at combined electromagnetic resonance and when the system is excited by a purely electric excitation (top-right, E) or a purely magnetic one (bottom-right, H). Color scale, incidence and polarization are preserved in all figures. and magnetic resonances overlap are determined by the geometrical parameters of the structure, as well as by the specific materials used. Therefore, it is possible to tune the wavelength of overlapping by an appropriate choice of these parameters. This gives the design a great degree of freedom, and makes it possible to operate at different frequencies. As an example, an Ag@Ge system with R=190 nm is presented (figures 2(e)–(h)). In this case, resonances overlap within 1500 and 1620 nm for Rin =50–60 nm. The particular case with Rin =55 nm is plotted in figure 2(h). In figure 3, the near-field pattern at the combined magnetic (shell) and electric (core) resonance is shown for the Ag@Si system. Also plotted are the responses of the system to purely electric and magnetic excitations. These were obtained by placing a perfect mirror at distances 3λ/4 and λ, respectively, from the center of the structure. The distinctive behavior of both contributions is preserved in the combined electromagnetic resonance, revealed through the rotating field confined within the Si shell (as in figure 1(e)), together with the dipolar LSPR resonance of the Ag core. Note that, indeed, the electric displacements in the core and the shell rotate in opposite directions and that the magnetic near-field pattern, figure 3(b), can be explained as a combination of the electric and magnetic contributions. Near-field patterns for core–shell with Ge covers are similar to those shown here. 3. Calculation of effective parameters What do we expect for a material consisting of such core–shell nanostructures? In general, it is highly nontrivial to extract the effective constitutive parameters of a metamaterial. Here, two different methods are applied to obtain the effective parameters of a metamaterial composed of the core–shell structures presented. In the first one, we will assume a random arrangement of the constituents, while in the second a metamaterial made of a cubic arrangement of them will be studied, both leading to negative-index behavior within certain wavelengths. New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 8 3.1. Effective parameters of a random arrangement of core–shell nanospheres For composites made of a cubic or random arrangement of dipolar particles, the Lorentz–Lorenz theory is widely used [11,28], leading to the well-known Clausius–Mossotti formulae relating the effective permittivity and permeability to the polarizabilities of the particles and the filling fraction f=(4/3)π N R3, where Nis the number of particles per unit volume: eff −0 eff + 20=fαE 4πR3,µeff −µ0 µeff + 2µ0=fαM 4πR3,(7) where αEand αMare the electric and magnetic polarizabilities of the spherical particles. For the core–shell structures considered R/λ ∼1/7. Therefore, we expect them to be well within the approximation considered in the theory, thus behaving essentially as electric and magnetic point dipoles. The electric and magnetic polarizabilities, αEand αM, respectively, are directly proportional to the scattering coefficients a1and b1(factor i(k3/6π)−1). In figures 4(a) and (b), we have plotted the polarizabilities for a core–shell configuration of Rin =47 nm and R=170 nm. It is clear from the graph that there is a spectral region where negative electric and magnetic polarizabilities are obtained. Interestingly, as mentioned before, no particular arrangement of the constituents is necessary to build the metamaterial, since the resonant electric and magnetic responses arise from each core–shell structure separately. Thus, assuming a random distribution we can compute the effective parameters from (7) for different filling fractions. Figures 4(c)–(e) depict the calculated effective permittivity, permeability and refractive index for metamaterials made up of these Ag@Si core–shells with filling fractions f=1/3, f=0.5 and f=2/3. The metamaterial has simultaneously negative permittivity and permeability for filling fractions higher than f=1/3. For a filling fraction of f=2/3, the system, moreover, has |neff|∼1, although this relatively high filling fraction would be in the limit of validity of Clausius–Mossotti formulae. In order to quantify the losses of the system, one can compute the so-called figure of merit, defined as f.o.m.=|<(neff)|/=(nneff). The computed f.o.m. in the LH spectral region for this filling fraction is indicated by the black curve, reaching a maximum value of f.o.m.∼0.71, corresponding to <(neff)∼−0.8. If Ge is used instead of Si, the NIM behavior starts with lower filling fractions, due to stronger electric and magnetic responses (see figure 5and compare the polarizabilities with those of figure 4). Superlensing capabilities are predicted for filling fractions lower than f=0.5 (figures 5(c)–(e)). Again, the f.o.m. for the highest filling fraction is plotted as a black curve. In this case, it reaches a maximum value of f.o.m.∼0.75, corresponding to <(neff)∼−0.5. Although far from being the best f.o.m. values reported for double-fishnet metamaterials (f.o.m.∼3)[7,29], the predicted values are reasonably good, with the obvious advantage of isotropy of this proposal. 3.2. Effective parameters of a cubic periodic arrangement of core–shell nanospheres To further test the metamaterial design, we consider now core–shell nanospheres arranged periodically at the vertices of a cubic lattice. We take germanium as the high-permittivity coating and choose the geometrical parameters as R=190 nm and Rin =55 nm. We consider the case when the cubic lattice has a period d=385 nm (corresponding to a filling fraction f∼0.5). Full numerical simulations (see section 5) of propagation across infinite slabs of different thicknesses ranging from 9 to 13 unit cells are carried out to ensure convergence. Bloch boundary conditions are applied in the interfaces with adjacent cells. Therefore, coupling between structures (with an New Journal of Physics 13 (2011) 123017 (http://www.njp.org/) 9 (a) (b) (c) (d) (e) Figure 4. (a–e) Electric and magnetic polarizabilities for an Ag@Si core–shell with Rin =47 nm and R=170 nm and effective parameters for a metamaterial composed of a random arrangement of these structures. (a) Real (red) and imaginary (blue) parts of the electric polarizability. (b) Real (red) and imaginary (blue) parts of the magnetic polarizability. (c–e) Real (red) and imaginary (blue) parts of the effective permittivity (c), permeability (d) and refractive index (e) of a metamaterial with several filling fractions: f=1/3 (dotted), f=0.5 (dashed) and f=2/3 (continuous). The black curve and the inset in (e) represent the f.o.m. for the highest filling fraction in the LH spectral region. interparticle distance of only 5 nm) is fully considered. The spectral range of the simulation is 1.28–1.7µm and the wave impinges at normal incidence. The effective material parameters are retrieved from the complex reflection and transmission coefficients through the usual equations found in the literature [20,21]. While the imaginary part is unambiguously computed from the simulation, once a sufficiently high number of unit cells in the propagation direction has been considered, the real part of the index of refraction must be determined with much more care. Concretely, one must pay special attention to ensure that the retrieved refractive index fulfills causality and passivity. In this way we have obtained two bounding values for the real part of the index, so it fulfills the two mentioned requirements. For a detailed description of how the index was obtained, see section 5. In figure 6, the obtained bounds for the refractive New Journal of Physics 13 (2011) 123017 (http://www.njp.org/)