## Abstract

The photonic band structures of superlattices composed of spatially-dispersive metal and polaritonic dielectric are theoretically investigated. The nonlocal relation between the electric current density and the electric field inside the metal layers is defined within the formalism of the Boltzmann kinetic equation, whereas the frequency dependent permittivity of the polar layers is modeled by a Lorentz-oscillator. Due to the large dielectric contrast between metal and polar components, the photonic band structure exhibits flat pass bands associated with Fabry-Perot resonances in the dielectric layers. There is also a wide stop band because of the existence of the polaritonic gap. We have compared our results with the predictions of the Drude-Lorentz model for the frequency-dependent metal permittivity. It is found that the nonlocal effect on the Fabry-Perot resonance bands is strong if their corresponding frequencies are within the interval where the difference between the impedances at both metal surfaces, predicted by the nonlocal and local formalisms, is maximal.

© 2015 Optical Society of America

## 1. Introduction

Periodic optical nanostructures with dielectric and metallic components belong to the meta-material systems because of their extraordinary electromagnetic properties (see, e.g., [1–9]). Among such properties, we should mention the negative refraction [10–12], which has been observed in double negative metamaterials, whose effective permittivity and permeability are simultaneously negative. The phenomenon of negative refraction is also observed in inherently-anisotropic dielectric-metal superlattices without negative effective permeability [13–15]. The optical properties of such multilayered metamaterials have been commonly described (see, e.g., [16–22] and references therein) within the Drude-Lorentz local model, which establishes a 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. The nonlocality (or spatial dispersion) is responsible for the anomalous skin effect in metals [23, 24]. Evidently, the simple Drude-Lorentz model absolutely ignores all these effects. Besides, 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.

In the paper [25] 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 a recent work [26], we have studied the photonic band structure for a one-dimensional periodic array of alternating dielectric and metal layers on the basis of the results obtained in [25]. As was shown in [26], the optic spectrum for a superlattice with very thin metallic layers exhibits narrow pass bands as a result of the strong contrast between the impedances of the dielectric and the metal. The narrow pass bands correspond 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. Noticeable differences not only in the magnitude, but also in the sign and the line-shape of the real part of the Bloch wave number in the Fabry-Perot resonance bands, were found.

The spatial dispersion in metallic layers is well manifested in the THz range [25,26]. Therefore, if a superlattice is composed of a nonlocal metal and a *resonant* dielectric (e.g. an ionic crystal or heteropolar semiconductor), the photonic band structure can be altered at frequencies of the forbidden band, which is created from the coupling between the transverse optical phonons and the transverse electromagnetic waves inside the dielectric. The effect of the polaritonic gap on the photonic band structure for periodic systems with solely dielectric components has been studied in several works (e.g. [27–39] and references therein). As is shown in [28], the photonic dispersion for *E*-polarized electromagnetic modes propagating in a two-dimensional (2D) photonic crystal, composed of materials with frequency-independent dielectric constants, exhibits gaps created from the periodicity of the lattice only. However, if one of the dielectric components is substituted by a resonant one, then a polaritonic gap appears in the photonic dispersion. Such a polaritonic gap turns out to be accompanied by two structural “twin” gaps because of the high dielectric contrast between components [28]. In [29, 31, 35], it was theoretically demonstrated that photonic crystals with phonon-polariton excitations exhibit a low-frequency pass band as well as flat bands directly below the transverse-optical phonon frequency *ω _{T}*, which are related to localized resonance modes in the polaritonic material having a high refractive index below

*ω*. Just above

_{T}*ω*, the polaritonic material possesses a very negative permittivity and behaves as metal, i.e. expels almost all of the field. In the latter case, the presence of pass bands above

_{T}*ω*can be correlated with the existence of bands in the photonic dispersion for a metallodielectric crystal at such frequencies. It is also interesting the appearance of defectlike photonic modes [35] in the polaritonic gap, below the longitudinal-optical phonon frequency

_{T}*ω*, where the refractive index of the polaritonic material is low. Also, photonic crystals composed of polar materials have been proposed for THz metamaterials. So, 2D composites composed of dielectric-polar or polar-polar material constituents behave like a bulk homogeneous uniaxial medium with hyperbolic dispersion relation in the THz range [40, 41]. Moreover, composites of two different kinds of non-overlapping spheres, one made from inherently non-magnetic polaritonic and the other from a Drude-like material, behave as double negative metamaterials at THz frequencies [42]. As is shown there, the polaritonic spheres are responsible for the negative effective magnetic permeability, whereas the Drude-like spheres give rise to a negative effective electric permittivity.

_{L}In the present work, we study the photonic band structure for a superlattice composed of alternating resonant dielectric and metal layers. Our approach is based on the Boltzmann kinetic equation for the distribution function of the conduction electrons, which allows the calculations of the general material equation, namely the nonlocal relation between the electrical current density and the electric field inside the metallic layer. As a result, the photonic dispersion relation for the resonant-dielectric/metal array is derived in terms of the surface impedances of the metal and resonant dielectric layers. We illustrate the nonlocal effect on the photonic band structure with calculations for a GaN-Al superlattice.

## 2. Problem formulation. Photonic dispersion relation

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

*d*, respectively. Thus, the size

_{b}*d*=

*d*+

_{a}*d*of a unit (

_{b}*a,b*) cell is the period of the bi-layer stack. The materials in the unit cell are assumed to be isotropic linear nonmagnetic media. Hence, the electric and magnetic components of an electromagnetic wave of frequency

*ω*, propagating along the

*x*axis (growth direction of the stack), can be oriented along the

*y*and

*z*axis, respectively:

All the *a*-layers are composed of a resonant dielectric, whose frequency-dependent dielectric function *ε _{a}*(

*ω*) is here modeled by using a Lorentz oscillator,

*ε*

_{∞}is the high-frequency dielectric constant determining the contribution of the valence electrons to the dielectric function,

*ω*and

_{TO}*ω*are respectively the transverse and longitudinal optical-phonon frequencies, and

_{LO}*ν*

_{0}denotes the damping constant. The model (2) defines refractive index

*n*, impedance

_{a}*Z*, wave number

_{a}*k*, and wave phase-shift

_{a}*φ*of the

_{a}*a*-layers,

*k*=

*ω/c*and

*c*being, respectively, the wave number and the velocity of light in vacuum. Within every

*a*-slab the electric field

*E*(

_{a}*x*) is governed by the Helmholtz equation, where the double prime denotes the double derivative with respect to

*x*. The magnetic field

*H*(

_{a}*x*) and derivative of the electric field within

*a*-slabs are related by Faraday law:

The general solution of Eqs. (4) and (5) for the *n*-th unit
(*a,b*) cell within the standing waves representation which is
the most suitable for our problem, reads

*a*-layer, where ${x}_{{a}_{n}}\u2a7d\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\u2a7d\phantom{\rule{0.2em}{0ex}}{x}_{{b}_{n}}$. The coordinates ${x}_{{a}_{n}}$ and ${x}_{{b}_{n}}$ refer to the left-hand edges of successive

_{n}*a*- and

_{n}*b*-layers, respectively. The integration constants in Eqs. (6) describe the magnetic field ${H}_{{a}_{n}}({x}_{{a}_{n}})$ at the left-hand and the magnetic field ${H}_{{a}_{n}}({x}_{{b}_{n}})$ on the right-hand edges of the

_{n}*a*-layer. Note that the thicknesses of individual layers are defined as

_{n}The Maxwell equation for the electric field distribution within the metallic *b*-slabs has the form

Here we have ignored the term *k*^{2}*E _{b}*(

*x*) originated from the displacement current since in metals it is negligibly small in comparison with the current of conduction electrons up to near-IR frequency range (see, e.g., [24]). Following article [25], the relation between the electron current density

*j*(

_{b}*x*) and the field

*E*(

_{b}*x*) in the metallic slab is obtained with the use of the Boltzmann kinetic equation. As a result, it gets the integral (i.e., nonlocal) form,

*x*is specified by the electric field at the same point,

Here *ω _{p}*,

*ν*and

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

_{F}*l*=

_{ω}*V*/(

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

To resolve Eq. (8) with the “non-local” current (9) inside the metallic *b _{n}*-layer, the electric field

*E*(

_{b}*x*) was suitably continued to the entire

*x*axis with the use of the parity and periodicity conditions (see [25] for details),

This gives rise to the presentation of the electromagnetic field as a Fourier series of the normal electromagnetic modes with discrete wave number *k _{s}* =

*πs/d*,

_{b}*b*-layer, where ${x}_{{b}_{n}}\phantom{\rule{0.2em}{0ex}}\u2a7d\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\u2a7d\phantom{\rule{0.2em}{0ex}}{x}_{{a}_{n+1}}$. The integration constants ${H}_{{b}_{n}}\left({x}_{{b}_{n}}\right)$ and ${H}_{{b}_{n}}\left({x}_{{a}_{n+1}}\right)$ represent the magnetic field at the corresponding layer edges. They are related to derivatives of the electric field by Faraday law which in the

_{n}*b*-layer reads

_{n}The response of the conduction electrons in metallic slabs to the *s*-th electromagnetic mode is specified by its own permittivity *ε*(*k _{s}*),

*not a metallic permittivity associated with the total electromagnetic field*. Note that in Eq. (14) the traditional one contributed by the displacement current omitted in Eq. (8), is absent. Indeed, one can make sure that $|\epsilon ({k}_{s})|\phantom{\rule{0.2em}{0ex}}\gg 1$ within all the frequency range up to near-IR. The mode permittivity

*ε*(

*k*) depends on the mode wave number

_{s}*k*via the

_{s}*nonlocality factor K*(

*k*),

_{s}l_{ω}Depending on the wave number *k _{s}*, the factor (15) entirely defines the spatial dispersion effect in the mode permittivity (14). Owing to this, it is worthwhile to write down its asymptotics,

These expressions show that the nonlocal effect can be negligible only when inequality (16a) is met for all the modes
contributing to the total electromagnetic field (12). Since in this case the nonlocality factor
*K* (*k _{s}l_{ω}*)

*≈*1, the permittivity

*ε*(

*k*) is the same for all the modes and coincides with that for the Drude-Lorentz model. As a consequence, the latter can be applied for the electrodynamic description of a metallic slab if provided by condition (16a).

_{s}The combination of Eqs. (6) and (12) with the continuity boundary conditions for the electric and magnetic fields taken at the interfaces
$x={x}_{{b}_{n}}$ and
$x={x}_{{a}_{n+1}}$, yields the recurrent relation describing the wave transfer through the whole *n*-th unit (*a,b*) cell,

The transfer matrix $\widehat{Q}$ has the following elements

Note that the determinant of
$\widehat{Q}$-matrix equals to unity, det
$\widehat{Q}$ = *Q*_{11}*Q*_{22} *− Q*_{12}*Q*_{21} = 1. For a periodic stack, the transfer *Q*^-matrix is independent of the cell index *n* since all the unit cells are identical.

As one can see, the transfer matrix is specified by the phase shift *φ _{a}* and the impedance

*Z*of the

_{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,

One should emphasize the different physical meaning of the introduced impedances: while *Z _{a}* is the surface impedance corresponding to a half-space (with only one surface), the surface impedances

*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 definitions (19) can be explicitly calculated resulting in

*b*-slabs: permittivity

*ε*=

_{b}*ε*(0), see (14), refractive index ${n}_{b}=\sqrt{{\epsilon}_{b}}$, impedance ${Z}_{b}={n}_{b}^{-1}$, wave number

*k*=

_{b}*kn*, and phase shift

_{b}*φ*=

_{b}*k*.

_{b}d_{b}To proceed further, we should address the Floquet theorem that can be properly written for the electromagnetic field at the left-hand edges
$x={x}_{{a}_{n+1}}$ and
$x={x}_{{a}_{n}}$ of *a _{n}*

_{+1}and

*a*layers,

_{n}*κ*traditionally standing for the Bloch wave number of a periodic structure. The unification of the transfer relation (17) with the Floquet theorem in the form (21) gives rise to the problem of eigenvectors and eigenvalues of the transfer $\widehat{Q}$-matrix. As a result (see details, e.g., in the book [20]), the desired dispersion relation for the Bloch wave number

*κ*of a one-dimensional periodic structure turns out to be determined by the trace of its unit-cell transfer matrix,

With the use of explicit expressions (18a) and (18d), one can readily obtain

Thus, the optic spectrum *κ*(*ω*) of the considered periodic bi-layer stack is defined by the universal dispersion relation (23). The difference of the kinetic approach from the Drude-Lorentz model emerges merely in the metallic impedances *ζ*_{0} and *ζ _{d}*. The transition from the kinetic approach to the Drude-Lorentz approximation is performed by the replacement of the nonlocality factor with one (

*K*(

*k*)

_{s}l_{ω}*→*1) for all of the summation indices

*s*in definitions (19), i.e. when the general expressions (19) can be described by their asymptotics (20). In the latter case the general dispersion relation (23) is reduced to its conventional counterpart valid for a wide class of bilayer stack-structures,

## 3. Analysis

In the case of a resonant-dielectric/metal superlattice, there exists a large contrast between the impedances of the *a*- and *b*-layers: the absolute value of the impedance *Z _{a}* is much greater than the absolute values of the metallic impedances

*ζ*

_{0}and

*ζ*,

_{d}*ω*. Therefore, the dispersion relation (23) for the photonic modes may have solutions for the Bloch wave number

_{TO}*κ*with

*|*Re

*κ < π/d*and 0 < Im

*κ < |*Re

*κ|*only in very narrow pass bands, associated with Fabry-Perot resonances arising in the

*a*-layer. Indeed, when

*|Z*

_{a}/ζ_{0}

*|*≫ 1 and

*|Z*≫ 1 in Eq. (23), such pass bands are close to the frequencies ${\omega}_{{j}_{a}}$ at which the condition

_{a}/ζ_{d}|It should be noted that the Drude-Lorentz impedances (20) 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*of the metallic slab thickness

_{b}/δ*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 (19) depend on the fourth control parameter

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

_{F}/c*K*(

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

_{s}l_{ω}(see details in [25]). For this reason, the nonlocal effect on the optic spectrum of photonic eigenmodes for the superlattice will be noticeable only in Fabry-Perot resonance bands appearing within such a frequency interval.

Below, we shall analyze the Fabry-Perot resonance bands of the photonic dispersion for a GaN-Al superlattice. In our numerical calculations, we use for the aluminium *b*-layers: *ν* = 0.00025*ω _{p}*, the Fermi velocity

*V*= 2.03 × 10

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

*ω*= 3.82 × 10

_{p}^{15}Hz,

*d*= 4

_{b}*δ*. The choice of GaN the

*a*-layers is due to the fact that its reststrahlen band (

*ω*⩽

_{TO}*ω*⩽

*ω*) is precisely within the interval (27) of strong Al nonlocality: the transverse and longitudinal phonon frequencies are respectively

_{LO}*ω*= 571cm

_{TO}

^{−}^{1}(17.12 THz) and

*ω*= 752cm

_{LO}

^{−}^{1}(22.54 THz) [43]. Other GaN parameters used here are:

*ε*

_{∞}= 8.9, and

*ν*

_{0}= 10

^{−}^{3}

*ω*. For the GaN-layer, we have chosen the thickness

_{TO}*d*such that the first Fabry-Perot resonance frequency

_{a}*ω*

_{1}

*≈*4

*ν*, i.e.

*d*=

_{a}*πc/ω*

_{1}Re

*n*(

_{a}*ω*

_{1}).

The normalized wave frequency *ω/ω _{p}* as a function of the wave number

*k*, which displays the polariton spectrum

_{a}*k*=

_{a}*kn*(

_{a}*ω*) for the GaN

*a*-layer, is depicted in Fig. 2. Note that the real part of

*k*(red lines) is much greater than the imaginary part (blue lines) everywhere except inside the GaN reststrahlen band (

_{a}*ω*⩽

_{TO}*ω*⩽

*ω*) wherein the real part is negligible with respect to the imaginary one. The dark dashed lines in Fig. 2 indicate frequencies

_{LO}*ω*corresponding to Fabry-Perot resonances (26). As is seen, for a fixed number

_{ja}*j*there are two resonance frequencies

_{a}*ω*, one of them is smaller than

_{ja}*ω*whereas the second one is above

_{TO}*ω*. Besides, the lower frequencies

_{LO}*ω*are accumulated just below the resonance frequency

_{ja}*ω*.

_{TO}Figure 3 exhibits first four of the Fabry-Perot resonance bands situated within the lower (THz) frequency range

The solid green and dashed red curves in Fig. 3 correspond
to the photonic dispersion predicted by the formalism of the Boltzmann kinetic
equation, whereas the dashed-dot blue and dashed-dot-dot black lines are obtained
within the Drude-Lorentz model. According to Fig.
3, the *j _{a}*-th resonance band appears below the
resonant frequency ${\omega}_{{j}_{a}}={j}_{a}\pi c/\mathrm{Re}{n}_{a}{d}_{a}$. Besides, the kinetic and local pass bands are
clearly distinguishable. Three main effects of spatial dispersion on the photonic
band structure can be observed in Fig. 3.
First of all, not only the magnitude but also the sign of real part
Re

*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 limited to the first Brillouin zone, are predicted in different Fabry-Perot resonance bands by the nonlocal and local models. Third, 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.

Figure 4 presents the first four Fabry-Perot resonance bands that are above the longitudinal optical phonon frequency *ω _{LO}*,

Note that the distinction between the predictions of the Boltzmann kinetic approach and the
Drude-Lorentz local model is not so strongly pronounced as for the first photonic
bands exhibited in Fig. 3. These results show
that the effect of the metal spatial dispersion on the photonic bands is well
manifest at the narrow Fabry-Perot resonances if the corresponding frequencies
*ω _{ja}* are within the interval where the
difference between the surface impedances

*ζ*

_{0}and

*ζ*, predicted by both nonlocal and local formalisms, is maximal. As is shown in [26] (Fig. 2 therein), in the case of Al-layers with

_{d}*d*= 4

_{b}*δ*, such a difference is considerable within the interval 0.001

*ω*≲

_{P}*ω*≲ 0.005

*ω*.

_{P}## 4. Conclusion

We have studied the photonic band structure of superlattices composed of nonlocal-metallic and resonant-dielectric layers. The study is based on the semiclassical kinetic approach of the Boltzmann kinetic equation, which allows us to calculate the nonlocal (integral) relation between the electric current density and the electric field in the metal layers. Besides, we have used a Lorentz-oscillator model for describing the resonant dielectric response of the polar layers. Because of the large dielectric contrast between metal and resonant-dielectric constituents, the photonic band structure is characterized by flat Fabry-Perot resonance bands. On the other hand, the existence of the polaritonic gap leads to the appearance of a wide stop band in the photonic band structure of the superlattice. Directly below the resonance frequency of the transverse optical phonons, a high concentration of flat Fabry-Perot resonance bands is observed. Unlike the dielectric-polar superlattices, the metal-dielectric heterostructure does not exhibit a low-frequency pass band for the electromagnetic modes propagating along the growth direction. Having compared our results with those obtained within the local Drude-Lorentz model for the metal permittivity, we found that the metal spatial dispersion noticeably alters the flat Fabry-Perot resonance bands for the photonic modes. It has been shown that the nonlocal effect is particularly important if the Fabry-Perot resonances are within the interval where the difference between the impedances at both metal surfaces, predicted by the nonlocal and local formalisms, is maximal.

## Acknowledgments

This work was partially supported by SEP-CONACYT (grant CB-2011-01-166382).

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