## Abstract

On the basis of the formalism of the Boltzmann kinetic equation for the distribution function of the conduction electrons, the photonic band structure of binary dielectric-metal superlattice is theoretically studied. Using the constitutive nonlocal relation between the electrical current density and the electric field inside the metallic layer, the dispersion equation for photonic eigenmodes in the periodic stack is analytically expressed in terms of the surface impedances at the interfaces of the metal and dielectric layers. In the case of very thin metallic layers, the optic spectrum for the superlattice exhibits narrow pass bands as a result of the strong contrast between the impedances of the dielectric and the metal. The narrow pass bands are attributed to Fabry-Perot resonances in the relatively-thick dielectric layer. The metal nonlocality is well pronounced in the infrared and, therefore, the nonlocal effect upon the photonic band structure of the superlattice can be strong when the Fabry-Perot resonance bands are in that frequency range. Our results for the photonic spectrum have been compared with those obtained within the local Drude-Lorentz model. Noticeably differences not only in the the magnitude, but also in the sign of the real part of the Bloch wave number in the Fabry-Perot resonance bands, have been found.

© 2014 Optical Society of America

## 1. Introduction

The Drude-Lorentz model is commonly used in the recent study of electrodynamics of metals. This model is a direct consequence of the local relation between the electron current density and the electric field. However, due to high mobility of conduction electrons and inhomogeneity of the electromagnetic field in metals, this relation actually takes an integral form and therefore, is nonlocal. As is known such a nonlocality gives rise to the effect of spatial dispersion, which is manifested in the anomalous skin effect. It is clear that the simple Drude-Lorentz model completely ignores all these effects. Moreover, in modern micro- and nanostructures, in which the size of the metallic inclusions is comparable with the skin depth of the electromagnetic field penetration, it is necessary to take into account the size effect. The skin effect (normal, anomalous, and infrared) described in all monographs on the physics of metals (see, e.g., [1,2]) was studied only in the bulk metals, i.e. for the model of a metallic half-space.

In [3] the electrodynamic problem for a metallic slab was solved analytically self-consistently including the nonlocal effect in the high-frequency conductivity of electrons. The current density was calculated within the semiclassical approach of the Boltzmann kinetic equation. In contrast to the Drude-Lorentz model, the kinetic approach gives rise to a nonlocal (integral) relation between the current density and the electric field. The distribution of the electromagnetic field inside the metallic slab was obtained in a closed analytical form. It has been shown that, in general, the results are qualitatively different from those obtained in the Drude-Lorentz approximation. In particular, in the high-frequency region (including the terahertz and infrared frequency range), the absorption oscillates with the radiation frequency and sample thickness. Also the absorption becomes sensitive to the Fermi velocity of electrons and depends nontrivially on the electron relaxation rate.

In the present work we study the photonic band structure for a one-dimensional periodic array of alternating dielectric and metal layers basing on the results of [3]. Such periodic systems (also known as multilayered metamaterials) are of great interest (see, e.g., [4–8] and references therein), however their optical properties are commonly described within the Drude-Lorentz model and, therefore, should be revised. Due to the remarkable progress in manufacturing one-dimensional heterostructures with given optical characteristics in the terahertz and/or infrared frequency range, a rigorous analysis gets an increasing importance. Within the Boltzmann kinetic approach that properly takes into account the spatial dispersion and the size effect, the photonic dispersion relation for the dielectric-metal superlattice is derived in its general form and compared with that obtained within the local Drude-Lorentz model.

## 2. Problem Formulation: Basic Relations

We consider an array of two alternating dielectric *a*- and metallic *b*-layers, see Fig. 1. Every kind of slabs has the constant thickness, *d _{a}* or

*d*, respectively. Thus, the size

_{b}*d*=

*d*+

_{a}*d*of a unit (

_{b}*a*,

*b*) cell is also the period of the bi-layer stack. The dielectric

*a*-slabs are specified by permittivity

*ε*, permeability

_{a}*μ*, corresponding refractive index ${n}_{a}=\sqrt{{\epsilon}_{a}{\mu}_{a}}$, impedance

_{a}*Z*=

_{a}*μ*/

_{a}*n*, wave number

_{a}*k*=

_{a}*n*and wave phase shift

_{a}k*φ*=

_{a}*k*(

_{a}d_{a}*k*=

*ω*/

*c*). We assume the permeability of the metallic

*b*-layers to be constant

*μ*, their other optic parameters will be introduced below.

_{b}The electromagnetic wave of frequency *ω* propagates perpendicularly to the stack with the electric and magnetic components as displayed in Fig. 1, **E**(*x*,*t*) = {0,*E*(*x*),0} exp(−*iωt*), **H**(*x*,*t*) = {0,0,*H*(*x*)} exp(−*iωt*). In such a geometry, within every *a*- or *b*-layer the relation between the electric *E*(*x*) and magnetic *H*(*x*) fields reads *ikμ _{a}*

_{,}

_{b}H_{a}_{,}

*(*

_{b}*x*) =

*E′*

_{a}_{,}

*(*

_{b}*x*), where the prime implies the derivative with respect to

*x*.

Within dielectric *a*-slabs the electric field *E _{a}*(

*x*) obeys the one-dimensional Helmholtz equation. Its general solution for the

*n*-th unit (

*a*,

*b*) cell can be presented as a superposition of forward and backward traveling plane waves,

*a*-layer, where

_{n}*x*

_{an}⩽

*x*⩽

*x*

_{bn}. Here ${A}_{n}^{\pm}$ are the complex amplitudes of the forward (+) and backward (−) traveling waves. The coordinates

*x*

_{an}and

*x*

_{bn}refer to the left-hand edges of successive

*a*- and

_{n}*b*-layers, respectively. The thicknesses of individual layers are defined as

_{n}*x*

_{bn}

*− x*

_{an}=

*d*and

_{a}*x*

_{an+1}−

*x*

_{bn}=

*d*. In accordance with the results of [3], the distribution of the electric field

_{b}*E*(

_{b}*x*) relevant to the nonlocal effect is given by

*b*-layer, where

_{n}*x*

_{bn}⩽

*x*⩽

*x*

_{an+1}. Expression (1b) represents a Fourier series of the normal electromagnetic modes with discrete wave number

*k*=

_{s}*πs*/

*d*. The interaction of

_{b}*s*-th mode with the conduction electrons of metallic slabs is specified by its own permittivity

*ε*(

*k*),

_{s}*not a permittivity associated with the total electromagnetic field*. The mode permittivity

*ε*(

*k*) depends on the mode wave number

_{s}*k*via the

_{s}*nonlocality factor 𝒦*(

*k*),

_{s}l_{ω}*ω*,

_{p}*ν*and

*V*are, respectively, the plasma frequency, relaxation rate and the Fermi velocity of the electrons;

_{F}*l*=

_{ω}*V*/(

_{F}*ν − iω*) implies the effective mean free path of the electrons due both to their collisions with scatters and to the phase change of the electromagnetic field.

Depending on the wave number *k _{s}*, the factor (3) entirely defines the spatial dispersion effect in the mode permittivity (2). Owing to this, it is worthwhile to write down its asymptotics,

*ε*(

*k*) is the same for all the modes and coincides with that for the Drude-Lorentz model, the latter can be applied for the electrodynamic description of a metallic slab.

_{s}The combination of Eq. (1) with the continuity boundary conditions for the electric and magnetic fields taken at the interfaces *x* = *x*_{bn} and *x* = *x*_{an+1}, yields the recurrent relation describing the wave transfer through the whole *n*-th unit (*a*,*b*) cell,

*Q̂*has the following elements

*Q̂*-matrix equals to unit, det

*Q̂*= 1. For a periodic stack, the transfer

*Q̂*-matrix is independent of the cell index

*n*since all the unit cells are identical.

The transfer matrix (6) is specified by the phase shift *φ _{a}* and the impedance

*Z*of the dielectric

_{a}*a*-layer, as well as, by the surface impedances

*ζ*

_{0}and

*ζ*of the left-hand and right-hand boundaries of the metallic

_{d}*b*-slab,

*Z*is the surface impedance corresponding to a half-space (with only one surface), the surfaces impedances

_{a}*ζ*

_{0}and

*ζ*inherently belong to a layer, thus, taking into account both waves, incident onto and reflected from a given surface. Within the Drude-Lorentz model the nonlocality factor

_{d}*𝒦*= 1 for all normal modes, and the sums in Eq. (7) can be explicitly calculated resulting in

*b*-slabs: impedance

*Z*=

_{b}*μ*/

_{b}*n*, phase shift

_{b}*φ*=

_{b}*k*, wave number

_{b}d_{b}*k*=

_{b}*n*, refractive index ${n}_{b}=\sqrt{{\epsilon}_{b}{\mu}_{b}}$ and permittivity

_{b}k*ε*=

_{b}*ε*(0), see Eq. (2).

As is known (see, e.g. the book [6]), the desired dispersion relation for the Bloch wave number *κ* of a one-dimensional periodic structure is determined by the trace of its unit-cell transfer matrix, 2cos(*κd*) = *Q*_{11} + *Q*_{22}. With the use of Eq. (6), one can readily obtain

*κ*(

*ω*) of the dielectric-metal periodic stack is defined by the universal dispersion relation (9). The difference of the kinetic approach from the Drude-Lorentz model emerges merely in the metallic impedances

*ζ*

_{0}and

*ζ*. The transition from the kinetic approach to the Drude-Lorentz approximation is performed by the replacement

_{d}*𝒦*(

*k*) → 1 for all of the summation indices

_{s}l_{ω}*s*in Eq. (7), i.e. when the general expressions (7) can be properly described by their asymptotics (8). In the latter case Eq. (9) degenerates into the conventional dispersion relation valid for a wide class of dielectric bilayer stack-structures.

## 3. Analysis

Because of the strong contrast between the dielectric, *Z _{a}*, and metallic,

*ζ*

_{0}and

*ζ*, impedances, the dispersion Eq. (9) for the photonic modes can have solutions for the Bloch wave number

_{d}*κ*with |Re(

*κ*)| <

*π*/

*d*and 0 < Im(

*κ*) < |Re(

*κ*)| only in pass bands associated with Fabry-Perot resonances emerging in the dielectric

*a*-layer. Indeed, when |

*Z*/

_{a}*ζ*

_{0}| ≫ 1 and |

*Z*/

_{a}*ζ*| ≫ 1 in Eq. (9), the solutions with minimal values of Im

_{d}*κ*are found in very narrow bands, being close to the frequencies

*ω*at which the Fabry-Perot resonance condition

_{j}*φ*=

_{a}*jπ*(

*j*= 1,2,3,...) is fulfilled.

It should be noted that the Drude-Lorentz impedances (8) depend on three dimensionless parameters: the frequency of the electromagnetic field *ω*/*ω _{p}* and the electron relaxation rate

*ν*/

*ω*normalized to the plasma frequency

_{p}*ω*, as well as the ratio

_{p}*d*/

_{b}*δ*of the metallic slab thickness

*d*to the minimum skin depth

_{b}*δ*=

*c*/

*ω*in the bulk metal, which is reached in the high-frequency range

_{p}*ν*≪

*ω*≪

*ω*, where ${\epsilon}_{b}=\epsilon (0)\approx -{\omega}_{p}^{2}/{\omega}^{2}$. It is of crucial importance that the kinetic impedances (7) depend on the fourth control parameter

_{p}*πV*/

_{F}*c*≪ 1 associated with the Fermi velocity of electrons. This parameter enters the argument of the nonlocality factor

*𝒦*(

*k*) and is responsible for the spatial dispersion effect, which is well manifested in the infrared if

_{s}l_{ω}*ν*<

*ω*< (

*πV*/

_{F}*c*)

*ω*(see details in [3]). For this reason, the nonlocal effect on the optic spectrum of photonic eigenmodes for the dielectric-metal superlattice will be noticeable only in Fabry-Perot resonance bands appearing in such a frequency interval.

_{p}As to the optic parameters of the dielectric *a*-layers, our results are valid not only when *ε _{a}* and

*μ*are positive constants. In general, they can be of complex values and/or frequency dispersive. However, for simplicity, we restrict our further analysis to the consideration of a vacuum-aluminum superlattice.

_{a}Figure 2 exhibits the frequency dependence of the surface impedances *ζ*_{0} and *ζ _{d}*, Eq. (7), used for calculating the photonic bands from Eq. (9). The metal parameters are:

*d*= 4

_{b}*δ*,

*V*= 2.03 × 10

_{F}^{8}cm/s,

*ω*= 3.82 × 10

_{p}^{15}s

^{−}^{1}and

*ν*= 0.00025

*ω*. As seen in Fig. 2, the differences of

_{p}*ζ*

_{0}(

*ω*) and

*ζ*(

_{d}*ω*) from those predicted by the local Drude-Lorentz model are of the order of 10

^{−4}that is the order of the real parts of the impedances themselves. In Fig. 3, we present the four lower Fabry-Perot resonance bands (panels a–d) for a vacuum-aluminum superlattice, whose

*a*-layer in the unit (

*a*,

*b*) cell has a thickness

*d*such that the frequency

_{a}*ω*

_{1}=

*πc*/

*d*for the first Fabry-Perot resonance in the

_{a}*a*-layer coincides with the frequency

*ω*= 9.708×10

^{−4}

*ω*of the absolute minimum of the difference $\text{Im}{\zeta}_{0}-\text{Im}{\zeta}_{0}^{(DL)}$ [red line in Fig. 2(c)]. The green and red curves in Fig. 3 correspond to the optic spectrum predicted by the kinetic approach, whereas the blue and black lines are obtained within the Drude-Lorentz model. According to Fig. 3, the

_{p}*j*-th resonance band appears below the frequency

*ω*=

_{j}*jπc*/

*d*. Besides, one can observe various effects of spatial dispersion on the photonic band structure. First, the kinetic and local pass bands are clearly distinguishable. Both the magnitude and the sign of real part Re

_{a}*κ*of the Bloch wave number can change in comparison with the predictions of the local model. Second, the jumps of Re

*κ*, which occur because Re

*κ*is confined to the first Brillouin zone, are different for the nonlocal and local models. Finally, the minimum value of the imaginary part Im

*κ*in the resonance bands practically does not vary from band to band within the nonlocal formalism.

## 4. Conclusion

We have studied the effect of spatial dispersion on the photonic band structure for dielectric-metal periodic superlattice, which is characterized by narrow Fabry-Perot resonance bands. We found that the nonlocality of the metal considerably alters the photonic spectrum in such bands. In particular, not only the magnitude, but also the sign of the real part of the Bloch wave number can differ from those predicted by the local Drude-Lorentz model.

## Acknowledgments

This work was partially supported by SEP-CONACYT (Mexico) under grant No. CB-2011-01-166382.

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