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Effective electronic response of a system of metallic cylinders

The electronic response of a composite consisting of aligned metallic cylinders in vacuum is investigated, on the basis of photonic band structure calculations. The effective long-wavelength dielectric response function is computed, as a function of the filling fraction. A spectral representation of the effective response is considered, and the surface mode strengths and positions are analyzed. The range of validity of a Maxwell-Garnett-like approach is discussed, and the impact of our results on absorption spectra and electron energy-loss phenomena is addressed.


I. INTRODUCTION
Both theoretical and experimental investigations of the optical properties of composite materials have been of basic importance over the years 1 . In particular, effective medium theories, in which the electromagnetic response of the composite is described in terms of an effective dielectric function ǫ eff , have been a useful tool for the interpretation of absorption spectra 2 . Mean-field theories of the effective response function have also been proved to be useful to analyze electron energy-loss experiments in which swift electrons penetrate composites at random distances from the particles 3−9 .
In the long-wavelength limit, the effective transverse dielectric function for a system of spherical particles was first derived by Maxwell-Garnett (MG) 10 , within a mean-field approximation valid for small values of the volume occupied by the spheres. A generalization of the MG theory to finite wavelength, taking account of the spatial nonlocality in the response, has been recently presented by Barrera and Fuchs 7 . On the other hand, many attempts have been made to account, at large filling fractions, for higher multipole interactions, which are absent in the MG theory 11−15 . Recently, exact numerical calculations for periodic dielectric structures have been performed 16 , based in the long-wavelength limit of photonic band structure calculations, and the MG theory has been shown to offer a good approximation in the low-filling-ratio regime. Finally, the discovery of tubular fullerenes 17 , a few years ago, has opened a new focus of interest for the application of these theories to the case of cylindrical interfaces 18,19 .
In this paper we evaluate, on the basis of photonic band structure calculations, the long-wavelength limit of the effective transverse dielectric function of a composite made up of long metallic cylinders embedded in an otherwise homogeneous medium. We first demonstrate that the composite can be replaced by an effective homogeneous medium, and then we determine the transverse dielectric function of that medium from the electromagnetic modes supported by such a homogeneous system. We consider a spectral representation of the effective dielectric function, we present an evaluation of the parameters involved, and we discuss the range of validity of a MG-like approach appropriate for cylindrical particles.
We also address the impact of our results on absorption spectra and electron energy-loss phenomena.

II. THEORY
It was shown by Bergman 21 and Milton 22 that in the long-wavelength limit the effective dielectric function of any macroscopically homogeneous two-component system of dielectric constants ǫ and ǫ 0 and volume fractions f and 1 − f , respectively, can be expressed as a sum of simple poles which depend only on the microgeometry of the composite material and not on the dielectric functions of the components: where u is the spectral variable m ν are depolarization factors, and B ν are the strengths of the corresponding normal modes, which all add up to unity: Similarly 23 , and ν C ν = 1.
In particular, if there is only one mode with a strength different from zero, one finds: and We consider a binary composite with a volume fraction 1 − f of insulator, with real dielectric constant ǫ 0 , and f of a periodic system of long metallic cylinders of diameter d arranged in a square array with lattice constant a = x d (See Fig. 1). The diameter of the cylinders is taken to be small in comparison with the wavelength of the electromagnetic excitation and large enough that a macroscopic dielectric function ǫ(ω) is ascribable to the cylinders. For simplicity, the magnetic permeabilities are assumed to be equal to unity in both media.
Now we take an electromagnetic (EM) wave normally incident on the structure, so that where E out represents the electric field intensity in the medium outside the cylinder. The effective dielectric function of isolated cylinders is then found to be given, again, by Eqs. The interaction between the particles in a composite has been considered, over the years, within the well-known MG approximation. The basic assumption of this approach is that the average electric field within a particle located in a system of identical particles is related to the average field in the medium outside as in the case of a single isolated particle, thus only dipole interactions being taken into account. Hence, after replacing, for p polarization, < E > being the macroscopic effective electric field in the composite, the MG result for cylindrical inclusions is found, which can be represented by the use of Eqs. (6) and (7) with and This is a special case of the generalized MG dielectric function derived in Ref. 25 for ellipsoidal inclusions, which in the case of aligned spheroids distributed at random has been extended beyond the MG approximation 26 . Higher multipole interactions are neglected in these approaches.
In order to compute, with full inclusion of higher multipoles, the effective dielectric function of our periodic system, we first follow Ref. 20 to calculate the photonic band structure.
We fix the frequency, the local dielectric function inside the cylinders being therefore specified, and we discretize the Maxwell equations within the unit cell. The discretized Maxwell equations provide a relationship between the electromagnetic fields on either side of the unit cell, i.e., z and z + a (see Fig. 1), and we exploit this to calculate, on applying Bloch's theorem, the eigenvalues of the so-called transfer matrix, which will give the band structure of the system.
Homogeneous media are well known to support only two transverse modes. On the other hand, looking at the imaginary part of the eigenvalues of the system, we observe that over the entire range of frequencies there are, for each polarization, two degenerate EM Bloch waves with vectors k and −k for which k(ω) roughly follows the dispersion relation of free light and which are dominant in the sense that they present the smallest attenuation (smallest imaginary part of k). Thus, we conclude that the composite can be replaced by an effective homogeneous medium, and we use the dispersion relation of these Bloch waves to compute the effective transverse dielectric function from where c is the speed of light.

III. RESULTS AND DISCUSSION
We have taken the insulating component of the composite to have a real dielectric constant of ǫ 0 = 1, and we have modeled the dielectric properties of the metal inside the cylinders by a simple free-electron (Drude) dielectric function: where ω p is the bulk plasma frequency of the metal, and γ represents an inverse electron relaxation time. We have used the plasma frequency of the conduction electrons in Aluminum, hω p = 15.8eV, and γ = 1.4eV.
For s polarized electromagnetic waves, our numerical results for the effective dielectric function, as obtained from Eq. (12), accurately reproduce the exact results of Eq. (6) and (7) with m = 0 and n = f , and this represents, therefore, a good check of our scheme.
In the case of p polarized electromagnetic excitations, great care was exercised to ensure In the case of a broad beam of charged particles penetrating the composite, the probability per unit path length, per unit energy, for the projectile to transfer energyhω to the medium is 29 where v and ǫ −1 ef f (q, ω) being the velocity of the incident particles and the effective inverse longitudinal dielectric function of the composite, respectively.hQ represents the component of the momentum transfer,hq, located in a plane perpendicular to the incident beam direction, Q c is determined by q 2 c = Q 2 c + q 2 v , q c representing the largest wave vector for which plasmons are well-defined excitations 30 , and In typical electron energy-loss experiments swift electrons of about 100keV penetrate the composite, and the main contribution to the integral of Eq. (15) comes, therefore, from very small values of the momentum transfer. Also, for small values of the adimensional parameter qa (qa < 1), a being the radius of the cylinder, the effective dielectric function of isolated cylinders has been proved to be rather insensitive to the introduction of a finite wave vector 9,29 , and the energy-loss spectra of swift electrons is therefore expected to be, for small values of the radius of the cylinders (a < 1nm) 31 , well described by the q → 0 limit of the imaginary part of the effective inverse dielectric function 32 .
Eq. (7) with n = 0 (dilute limit of n = f ) and n = 1/2 exactly coincides, in the longwavelength limit, with the result one obtains for the effective inverse longitudinal dielectric function of isolated cylinders when the momentum transfer is parallel to the axis and when it is located in a plane perpendicular to the axis of the cylinder, respectively 29 . In this limit (q → 0) bulk plasmons are not excited, as a consequence of the so-called Bregenzung effect, and the pole in Eq. (7) at ǫ(ω) + ǫ 0 = 0, i.e., ω 1 = √ m 1 ω p with m 1 = 1/2, determines, for p polarization, the frequency of the so-called surface plasmon resonance at which electron energy-loss occurs. Fig. 4 shows, as a function of the frequency, our calculations of Im[−ǫ −1 eff (ω)]/f obtained for various filling fractions by varying the ratio x between the lattice constant and the diameter of the cylinders. In a homogeneous metal the so-called energy-loss function, Im[−ǫ −1 eff (ω)], shows a single peak at the bulk plasmon resonance, and this peak persists in the composite for x ≤ 1, i.e., for filling fractions of metal for which the insulator no longer forms a connected medium. Besides this peak, there is also a band in Im[−ǫ −1 eff (ω)] from multipole contributions to the effective response at ω ν = n ν (f )ω p (ν = 2, ... ; n ν < n 1 ). As in the case of Im [ǫ eff (ω)], the energy-loss function is well described by the MG approximation in the low-filling-ratio regime, and our results begin to deviate from those obtained within the MG approximation at x ≈ 2 (see Fig. 5). We also find that the depolarization factors n 1 (f ) are larger than the MG depolarization factor of Eq. (11) for all volume fractions of metal for which the MG theory does not reproduce our results.
Our numerical calculations of the effective dielectric function, as obtained from Eq. (12), lead us to the conclusion that the depolarization factors m 1 and n 1 of Eqs. (1) and (4), corresponding to dipolar modes, satisfy the following relation:  6) and (7) with the depolarization factors m 1 and n 1 given by Eqs. (10) and (11), i.e., the MG approximation.
On the other hand, as long as multipolar modes give non-negligible contributions to the spectral representation of the effective response, i.e., B ν = 0 and C ν = 0 (ν = 2, ...), the strengths of the dipolar modes, B 1 and C 1 , become smaller than unity (see Eqs. (3) and (5)), and a combination of Eqs. (1) and (4)  and we have thus represented in Fig. 6, in the case of p polarization, universal curves for the actual dipolar mode depolarization factors and strengths versus the filling fraction. It is obvious from this figure that the trend with increasing filling fraction is for the dipolar peaks in Im [ǫ eff (ω)] / Im −ǫ −1 eff (ω) to move from the cylinder dipole mode (m 1 = n 1 = 1/2) to lower / higher energies. Notice that the MG results, also plotted in this figure, agree with our numerical results for f < 0.2 (x > 2). Nevertheless, at higher filling fractions the actual depolarization factors approach more rapidly, as f is increased, the depolarization factors appropriate to the homogeneous metal (m 1 = 0 and n 1 = 1). In particular, for x < 1 (f > π/4) n 1 = 1, indicating that losses due to the bulk plasmon are expected when the metal forms a connected medium.
Associated to the deviation of the actual dipolar mode positions from the MG results is the appearance of non-negligible multipolar mode strengths, B ν (ν = 2, ...) and C ν (ν = 2, ...), and, accordingly (see Eqs. (3) and (5)), a reduction in the dipolar mode strengths In summary, we have evaluated, on the basis of photonic band structure calculations, the effective long-wavelength dielectric response function of a system of long metallic cylinders.
We have considered a spectral representation of both direct and inverse effective dielectric functions, of interest in the interpretation of absorption spectra and electron energy-loss experiments, respectively. We have analyzed the surface mode strengths and positions, and the effect of the electromagnetic interactions between the cylinders has been investigated.
We have concluded that MG results are good as long as the distance between the axis of neighbouring cylinders is larger than twice the diameter of the cylinders. This is in contrast with the results we have also obtained for long dielectric cylinders. In this case we have found the MG results to be good for almost all the filling ratios for which the cylinders are not touching, as happens in the case of a periodic array of dielectric spheres 16 . Mode strengths