THz photonic bands of periodic stacks composed of resonant dielectric and nonlocal metal

Fernando Díaz-Monge; Alejandro Paredes-Juárez; Denis A. Iakushev; Nykolay M. Makarov; Felipe Pérez-Rodríguez

doi:10.1364/OME.5.000361

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_T. Just above ω_T, 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 ω_T 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 ω_L, 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.

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_b, respectively. Thus, the size d = d_a + d_b of a unit (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:

\begin{matrix} E (x, t) = {0, E (x), 0} \exp (- i ω t), \\ H (x, t) = {0, 0, H (x)} \exp (- i ω t) . \end{matrix}

Fig. 1 Scheme of the bi-layer stack

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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_{a} = ε_{\infty} (1 - \frac{ω_{L O}^{2} - ω_{T O}^{2}}{ω^{2} - ω_{T O}^{2} + i ω v_{0}}),

where ε_∞ is the high-frequency dielectric constant determining the contribution of the valence electrons to the dielectric function, ω_TO and ω_LO are respectively the transverse and longitudinal optical-phonon frequencies, and ν₀ denotes the damping constant. The model (2) defines refractive index n_a, impedance Z_a, wave number k_a, and wave phase-shift φ_a of the a-layers,

n_{a} = \sqrt{ε_{a}}, Z_{a} = n_{a}^{- 1}, k_{a} = k n_{a}, φ_{a} = k_{a} d_{a},

with 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,

E_{a}^{″} (x) + k_{a}^{2} E_{a} (x) = 0,

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:

i k H_{a} (x) = E_{a}^{'} (x) .

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

E_{a_{n}} (x) = i Z_{a} H_{a_{n}} (x_{a_{n}}) \frac{\cos [k_{a} (x_{b_{n}} - x)]}{\sin (k_{a} d_{a})} - i Z_{a} H_{a_{n}} (x_{b_{n}}) \frac{\cos [k_{a} (x - x_{a_{n}})]}{\sin (k_{a} d_{a})},

H_{a_{n}} (x) = H_{a_{n}} (x_{a_{n}}) \frac{\sin [k_{a} (x_{b_{n}} - x)]}{\sin (k_{a} d_{a})} + H_{a_{n}} (x_{b_{n}}) \frac{\sin [k_{a} (x - x_{a_{n}})]}{\sin (k_{a} d_{a})}

inside the a_n-layer, where

x_{a_{n}} ⩽ x ⩽ x_{b_{n}}

. The coordinates

x_{a_{n}}

and

x_{b_{n}}

refer to the left-hand edges of successive a_n - and b_n -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 a_n -layer. Note that the thicknesses of individual layers are defined as

x_{b_{n}} - x_{a_{n}} = d_{a}, x_{a_{n + 1}} - x_{b_{n}} = d_{b}, x_{a_{n + 1}} - x_{a_{n}} = d .

The Maxwell equation for the electric field distribution within the metallic b-slabs has the form

E_{b}^{″} (x) + \frac{4 π i k^{2}}{ω} j_{b} (x) = 0.

Here we have ignored the term k²E_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,

j_{b} (x) = \int_{- \infty}^{\infty} d x' σ (x - x') E_{b} (x'),

σ (x) = \frac{3 σ_{DL}}{4 l_{ω}} \int_{0}^{1} d n_{x} \frac{1 - n_{x}^{2}}{n_{x}} \exp (- \frac{| x |}{l_{ω} n_{x}}),

in contrast to the Drude-Lorentz model that is local by definition, i.e., in which the current density at the point x is specified by the electric field at the same point,

j_{b} (x) = σ_{DL} E_{b} (x), σ_{DL} = \frac{ω_{p}^{2}}{4 π (v - i ω)} .

Here ω_p, ν and V_F are, respectively, the plasma frequency, the relaxation rate and the Fermi velocity of the conduction electrons, 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),

E_{b_{n}} (x) = E_{b_{n}} (2 x_{b_{n}} - x), E_{b_{n}} (x) = E_{b_{n}} (x \pm 2 d_{b}) .

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,

\begin{array}{l} E_{b_{n}} (x) = - \frac{i k H_{b_{n}} (x_{b_{n}})}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{\cos [k_{s} (x - x_{b_{n}})]}{k_{s}^{2} - k^{2} ε (k_{s})} \\ + \frac{i k H_{b_{n}} (x_{a_{n + 1}})}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{\cos [k_{s} (x_{a_{n + 1}} - x)]}{k_{s}^{2} - k^{2} ε (k_{s})}, \end{array}

\begin{array}{l} H_{b_{n}} (x) = - \frac{H_{b_{n}} (x_{b_{n}})}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{k_{s} \sin [k_{s} (x - x_{b_{n}})]}{k_{s}^{2} - k^{2} ε (k_{s})} \\ + \frac{H_{b_{n}} (x_{a_{n + 1}})}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{k_{s} \sin [k_{s} (x_{a_{n + 1}} - x)]}{k_{s}^{2} - k^{2} ε (k_{s})}, \end{array}

inside the b_n-layer, where

x_{b_{n}} ⩽ x ⩽ x_{a_{n + 1}}

. The integration constants

H_{b_{n}} (x_{b_{n}})

and

H_{b_{n}} (x_{a_{n + 1}})

represent the magnetic field at the corresponding layer edges. They are related to derivatives of the electric field by Faraday law which in the b_n-layer reads

i k H_{b} (x) = E_{b}^{'} (x) .

The response of the conduction electrons in metallic slabs to the s-th electromagnetic mode is specified by its own permittivity ε(k_s),

ε (k_{s}) = - \frac{ω_{p}^{2}}{ω (ω + i v)} K (k_{s} l_{ω}),

which is 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

| ε (k_{s}) | ≫ 1

within all the frequency range up to near-IR. The mode permittivity ε(k_s) depends on the mode wave number k_s via the nonlocality factor K (k_sl_ω),

\begin{array}{l} K (k_{s} l_{ω}) = \frac{3}{2} \int_{0}^{1} \frac{(1 - n_{x}^{2}) d n_{x}}{1 + {(k_{s} l_{ω} n_{x})}^{2}} \\ = \frac{3}{2} {[\frac{1}{k_{s} l_{ω}} + \frac{1}{{(k_{s} l_{ω})}^{3}}] \arctan (k_{s} l_{ω}) - \frac{1}{{(k_{s} l_{ω})}^{2}}} . \end{array}

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,

K (k_{s} l_{ω}) \approx 1 - {(k_{s} l_{ω})}^{2} / 5, {(k_{s} | l_{ω} |)}^{2} ≪ 1;

K (k_{s} l_{ω}) \approx 3 π / 4 | k_{s} | l_{ω}, | k_{s} l_{ω} | ≫ 1.

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_sl_ω) ≈ 1, the permittivity ε(k_s) 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).

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,

(\begin{array}{l} E (x_{a_{n + 1}}) \\ H (x_{a_{n + 1}}) \end{array}) = (\begin{matrix} Q_{11} & Q_{12} \\ Q_{21} & Q_{22} \end{matrix}) (\begin{array}{l} E (x_{a_{n}}) \\ H (x_{a_{n}}) \end{array}) .

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

Q_{11} = \frac{ζ_{0}}{ζ_{d}} \cos φ_{a} - i \frac{ζ_{0}^{2} - ζ_{d}^{2}}{Z_{a} ζ_{d}} \sin φ_{a},

Q_{12} = \frac{ζ_{0}^{2} - ζ_{d}^{2}}{ζ_{d}} \cos φ_{a} + i \frac{Z_{a} ζ_{0}}{ζ_{d}} \sin φ_{a},

Q_{21} = - \frac{1}{ζ_{d}} \cos φ_{a} + i \frac{ζ_{0}}{Z_{a} ζ_{d}} \sin φ_{a},

Q_{22} = \frac{ζ_{0}}{ζ_{d}} \cos φ_{a} - i \frac{Z_{a}}{ζ_{d}} \sin φ_{a} .

Note that the determinant of $\hat{Q}$ -matrix equals to unity, det $\hat{Q}$ = Q₁₁Q₂₂ − Q₁₂Q₂₁ = 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_a of the a-layer, as well as, by the surface impedances ζ₀ and ζ_d of the left-hand and right-hand boundaries of the metallic b-slab,

ζ_{0} = - \frac{i k}{d_{b}} \sum_{s - \infty}^{\infty} \frac{1}{k_{s}^{2} - k^{2} ε (k_{s})}, ζ_{d} = - \frac{i k}{d_{b}} \sum_{s = - \infty}^{\infty} \frac{\cos (k_{s} d_{b})}{k_{s}^{2} - k^{2} ε (k_{s})} .

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 ζ₀ and ζ_d 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 K = 1 for all normal modes, and the sums in definitions (19) can be explicitly calculated resulting in

ζ_{0}^{(D L)} = i Z_{b} \frac{\cos φ_{b}}{\sin φ_{b}}, ζ_{d}^{(D L)} = i Z_{b} \frac{1}{\sin φ_{b}},

with the Drude-Lorentz optic parameters of metallic b-slabs: permittivity ε_b = ε(0), see (14), refractive index

n_{b} = \sqrt{ε_{b}}

, impedance

Z_{b} = n_{b}^{- 1}

, wave number k_b = kn_b, and phase shift φ_b = k_bd_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_n layers,

(\begin{matrix} E (x_{a_{n + 1}}) \\ H (x_{a_{n + 1}}) \end{matrix}) = (\begin{matrix} \exp (i κ d) & 0 \\ 0 & \exp (i κ d) \end{matrix}) (\begin{matrix} E (x_{a_{n}}) \\ H (x_{a_{n}}) \end{matrix}),

with 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

\hat{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,

2 \cos (κ d) = Q_{11} + Q_{22} .

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

\cos (κ d) = \frac{ζ_{0}}{ζ_{d}} \cos φ_{a} - i \frac{Z_{a}^{2} + ζ_{0}^{2} - ζ_{d}^{2}}{2 Z_{a} ζ_{d}} \sin φ_{a} .

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 ζ₀ 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_sl_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,

\cos (κ d) = \cos φ_{a} \cos φ_{b} - \frac{1}{2} (\frac{Z_{a}}{Z_{b}} + \frac{Z_{b}}{Z_{a}}) \sin φ_{a} \sin φ_{b} .

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 ζ₀ and ζ_d,

| Z_{a} | ≫ max (| ζ_{0} |, | ζ_{d} |),

even at frequencies very close to the resonant one, ω_TO. Therefore, the dispersion relation (23) for the photonic modes may have solutions for the Bloch wave number κ with |Reκ < π/d and 0 < Imκ < |Reκ| only in very narrow pass bands, associated with Fabry-Perot resonances arising in the a-layer. Indeed, when |Z_a/ζ₀| ≫ 1 and |Z_a/ζ_d| ≫ 1 in Eq. (23), such pass bands are close to the frequencies

ω_{j_{a}}

at which the condition

Re φ_{a} \equiv Re k n_{a} d_{a} = j_{a} π, j_{a} = 1, 2, 3 \dots .,

is satisfied.

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_p normalized to the plasma frequency ω_p, as well as the ratio d_b/δ of the metallic slab thickness d_b to the minimum skin depth δ = c/ω_p in the bulk metal, which is reached in the high-frequency range ν ≪ ω ≪ ω_p, where $ε_{b} = ε (0) \approx - ω_{p}^{2} / ω^{2}$ . It is of crucial importance that the kinetic impedances (19) depend on the fourth control parameter πV_F/c ≪ 1 associated with the Fermi velocity of electrons. This parameter enters the argument of the nonlocality factor K (k_sl_ω) and is responsible for the spatial dispersion effect, which is well manifested in the infrared if

ν < ω < π V_{F} ω_{p} / c

(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 = 0.00025ω_p, the Fermi velocity V_F = 2.03 × 10⁸cm/s, ω_p = 3.82 × 10¹⁵Hz, d_b = 4δ. The choice of GaN the a-layers is due to the fact that its reststrahlen band (ω_TO ⩽ ω ⩽ ω_LO) is precisely within the interval (27) of strong Al nonlocality: the transverse and longitudinal phonon frequencies are respectively ω_TO = 571cm⁻¹ (17.12 THz) and ω_LO = 752cm⁻¹ (22.54 THz) [43]. Other GaN parameters used here are: ε_∞ = 8.9, and ν₀ = 10⁻³ω_TO. For the GaN-layer, we have chosen the thickness d_a such that the first Fabry-Perot resonance frequency ω₁ ≈ 4ν, i.e. d_a = πc/ω₁Ren_a(ω₁).

The normalized wave frequency ω/ω_p as a function of the wave number k_a, which displays the polariton spectrum k_a = kn_a(ω) for the GaN a-layer, is depicted in Fig. 2. Note that the real part of k_a (red lines) is much greater than the imaginary part (blue lines) everywhere except inside the GaN reststrahlen band (ω_TO ⩽ ω ⩽ ω_LO) wherein the real part is negligible with respect to the imaginary one. The dark dashed lines in Fig. 2 indicate frequencies ω_ja corresponding to Fabry-Perot resonances (26). As is seen, for a fixed number j_a there are two resonance frequencies ω_ja, one of them is smaller than ω_TO whereas the second one is above ω_LO. Besides, the lower frequencies ω_ja are accumulated just below the resonance frequency ω_TO.

Fig. 2 Polaritonic spectrum of the GaN a-layer.

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

0 \underset{˜}{<} ω \underset{˜}{<} ω_{T O} .

Fig. 3 Photonic band structure in the vicinities of the four lower Fabry-Perot resonances for a GaN-Al superlattice, predicted by the nonlocal (Boltzmann) and local (Drude-Lorentz) formalisms.

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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 $ω_{j_{a}} = j_{a} π c / Re n_{a} d_{a}$ . 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 Reκ of the Bloch wave number can change in comparison with the predictions of the local model. Second, the jumps of Reκ, which occur because Reκ 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 Imκ 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,

ω_{L O} ≲ ω .

Fig. 4 Photonic band structure in the vicinities of the first four Fabry-Perot resonances above ω_LO for a GaN-Al superlattice as in Fig. 3.

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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 ζ₀ and ζ_d, 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_b = 4δ, such a difference is considerable within the interval 0.001ω_P ≲ ω ≲ 0.005ω_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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THz photonic bands of periodic stacks composed of resonant dielectric and nonlocal metal

Abstract

1. Introduction

2. Problem formulation. Photonic dispersion relation

3. Analysis

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (36)

Optical Materials Express