Electrodynamics of superlattices with ultra-thin metal layers: quantum Landau damping and band gaps with nonzero density of states

S. G. Castillo-López; A. A. Krokhin; N. M. Makarov; F. Pérez-Rodríguez

doi:10.1364/OME.9.000673

1. Introduction

Fermi sea of conduction electrons is a reliable protection of metal interior from electromagnetic fields. Due to high concentration of free electrons, the metals reflect the essential part of electromagnetic energy and the rest which penetrates inside is dissipated in a narrow skin layer of thickness δ ∼ 100 nm. Even metals with low electron mobility where electron mean free path l is less than δ serve as very good shields. Exponential decay of electromagnetic field inside metal in this case is referred to as normal skin-effect [1]. Increase of electron mobility, which occurs in metallic samples of high-quality and/or with lowing temperature require less electrons to produce the necessary currents in the skin layer and those electrons which move almost perpendicular to the metal surface become “ineffective.” In the extreme case, l ≫ δ, most of the electrons are ineffective and the skin layer is formed by a small group of electrons moving almost parallel to the surface [2]. The electromagnetic response of metal in this case of so-called anomalous skin-effect is different from that in the case normal skin-effect. In particular, the field inside metal decays slower than exponential and electromagnetic absorption remains finite even in the collisionless limit l → ∞ [3]. The latter “anomaly” is a general property of colisionless plasma known as Landau damping [4]. In metals this mechanism of energy absorption of propagating wave is manifested together with the nonlocal effects in metal conductivity since effective electrons move without relaxation a distance l, which exceeds the typical wave scale in metal δ. Traditionally the effect of Landau damping was associated with bulk metal samples. However, in the past few years, it was shown that Landau damping always exists and substantially alters the electrodynamics of nanometric metallic systems (see, e.g., Refs. [5–10] and references therein).

Apart from the anomalous skin-effect there are several other effects, known in pure metals, which are due to a specific small group of electrons, when the other electrons forming metal conductivity can be formally ignored. Here we would like to mention Azbel-Kaner cyclotron resonance when electrons orbiting in magnetic field periodically return back to the skin layer and absorb resonantly electromagnetic energy [11]. Two small groups of electrons are responsible for the cyclotron resonance. One group of electrons move inside a narrow skin layer, continuously absorbing energy through the mechanism of Landau damping. Another group moves along Larmor orbits periodically entering for a short time the skin layer where they are resonantly accelerated by ac electric field. The rest of the Larmor circle, which is much larger than δ, the electrons move freely since the electromagnetic field sharply decays beyond the skin layer. The electrons of the second group are the carriers of Azbel-Kaner cyclotron resonance.

Here we consider a resonance effect which occurs in ultra-thin metal films arranged in a periodic superlattice. In a thin metal film the transversal motion of conduction electrons is quantized because of multiple reflection from the metal surfaces, while the electrons move freely along the film. Quantum size-effect in single metal films, wires and nano-particles has been observed using electron tunneling [12], nuclear magnetic resonance [13], and dc measurements with scanning tunneling microscopy (STM) [14]. STM images of size-quantized electron states were observed in the MBE-grown lead quantum wedge [15].

Electron transitions between size-quantized levels occur with absorption/emission of photon. This mechanism is the quantized version of previously mentioned Landau damping in bulk samples. Similar to electron dynamics in Azbel-Kaner cyclotron resonance, there are two small groups of conduction electrons in an ultra-thin metal film. They interact differently with external electromagnetic wave. Electrons in one group moves practically parallel to the metal surface forming the skin layer. They give a nonlocal contribution to the electromagnetic response which can be described classically. Electrons of the other group execute quantum transitions between the states which are in resonance with external wave. They require quantum approach. We apply the linear response theory developed by Kubo to calculate nonlocal electron conductivity of a thin metal film.

When metal films are arranged in a metal-dielectric superlattice, one more resonance becomes possible – it is Fabry-Perot resonance arising in dielectric spacers between neighboring metal films. Due to this resonance relatively narrow pass bands exist in the transmission spectrum of a superlattice. If the two resonances, Fabry-Perot and quantized Landau damping, are close to each other, the effect of level repulsion takes place leading to essential changes in the band structure of a superlattice and in its transmission and absorption spectra. We demonstrate that depending on how close these resonances are, a new pass band or a new gap may appear in the photonic spectrum. The mechanism of formation of these new features is different from the conventional mechanism of Bragg reflection. In particular, the density of photonic states remains finite within the new gap but the transmission is suppressed due to the quantized Landau damping. The photonic states existing within the new gap are “dark” states with strongly anomalous dispersion. Such dark states have not been observed so far and their role in electrodynamics of metal-dielectric superlattices requires a detailed study.

2. Formulation of the problem

We study the electromagnetic response of a binary layered stack which is unilaterally excited by a plane wave of frequency ω. Each unit cell of size d = d_a + d_b is composed of a dielectric a-layer and a metallic b-film, with thicknesses d_a and d_b, respectively. The electromagnetic wave propagates along the growth direction of the superlattice (axis x), with polarization of the electric E(x, t) = {0, E (x), 0} exp(−iωt) and magnetic H(x, t) = {0, 0, H (x)} exp(−iωt) fields along y- and z-axis, as it is displayed in Fig. 1.

Fig. 1 Geometry of the binary periodic stack and propagating electromagnetic wave. The stack region II is embedded between media I and III.

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Inside the dielectric a-layer of the ℓth unit (a, b) cell (ℓ = 1, 2, 3, . . ., N) the electromagnetic field is represented by a superposition of plane waves:

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

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

where x_{a_ℓ} ⩽ x ⩽ x_{b_ℓ}. The coordinates x_{a_ℓ} and x_{b_ℓ} = x_{a_ℓ} + d_a refer to the left-hand edges of successive a_ℓ- and b_ℓ-layers, respectively, see Fig. 1. Note that x_{a_ℓ+1} − x_{a_ℓ} = d constitutes the size of unit (a, b) cell. The electromagnetic wave number in the lossless dielectric with permittivity ε_a is k

k_{a} = k \sqrt{ε_{a}}

(k = ω/c). The integration constants H_{a_ℓ} (x_{a_ℓ}) and H_{a_ℓ} (x_{b_ℓ}) describe, respectively, the magnetic field at the left- and right-hand edges of the a_ℓ-layer.

Because of strong spatial dispersion and quantum regime the electric and magnetic fields within the ℓth metal b-film have to be calculated using nonlocal approach based on quantum Kubo formalism. According to the results recently obtained in Refs. [9,10], these inhomogeneous fields can be written as the following Fourier series:

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

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

for x_{b_ℓ} ⩽ x ⩽ x_{a_ℓ+1} = x_{b_ℓ} +d_b. Each term in Eqs. (2) represents a discrete normal electromagnetic s-mode with quantized wave numbers k_s = sπ/d_b. Whereas under the condition of strong spatial dispersion, the permittivity of metal cannot be introduced [1], each normal s-mode contributes to the linear electromagnetic response with its proper k_s-dependent permittivity ε(k_s),

ε (k_{s}) = 1 - \frac{ω_{p}^{2}}{ω^{2}} 𝒬 (k_{s}) .

The factor 𝒬(k_s) in this formula is responsible for nonlocality of electron plasma and quantization of electron momentum,

𝒬 (k_{s}) = \frac{3 ω}{4} {(\frac{k_{F} d_{b}}{π})}^{- 1} {\sum_{n = - N_{F}}^{N_{F}}}^{'} \frac{1 - q_{n}^{2} / k_{F}^{2}}{ω - ω_{n + | s |, n} + i ν} .

Here the prime at the sum-symbol indicates that the term with n = 0 is omitted, ω_p implies the plasma frequency, and ν is regarded as the electrons relaxation rate due to their collisions with scatters in the metal film. The detailed calculations of the field distribution (2) as a superposition of discrete normal electromagnetic modes defined by the mode permittivity (3) with the quantum nonlocality factor (4) can be found in Ref. [10].

Inside the metallic layer the transversal electron wave number is quantized, q_x ≡ q_n = nπ/d_b. Since the total electron energy equals to Fermi energy ∊_F the number of open electron channels turns out to be finite, |n| ⩽ N_F = [k_Fd_b/π], where square brackets [. . .] denote the integer part of inner quantity and $k_{F} = \sqrt{2 m ∊_{F}} / ℏ$ is the Fermi wave number with m being the effective electron mass. Quantum transition between the n th and (n + |s|)th channels is accompanied by absorption (or emission) of photon with momentum ħ|k_s|. In view of the evident conservation law q_n+|s| = q_n + |k_s|, the corresponding transition frequency reads

ω_{n + | s |, n} = \frac{ℏ}{2 m} (q_{n + | s |}^{2} - q_{n}^{2}) = ω_{n, s} + ω_{s};

ω_{n, s} = \frac{ℏ | k_{s} | q_{n}}{m} = | k_{s} | V_{F} \frac{q_{n}}{k_{F}}, ω_{s} = \frac{ℏ k_{s}^{2}}{2 m},

where

V_{F} = \sqrt{2 ∊_{F} / m}

is the Fermi velocity of the conduction electrons. The quantum transition frequency ω_n+|s|,n, Eq. (5a), consists of two terms: ω_n,s represents the quasi-classical version of ω_n+|s|,n, whereas ω_s is known as the recoil frequency that does not have a classical counterpart. In other words, the total quantum transition frequency ω_n+|s|,n approaches ω_n,s in the quasi-classical limit since the recoil frequency ω_s vanishes.

The equations (2) – (4) predict the resonant behavior of the electromagnetic response of metallic nano-thin films. Each of normal |s|-mode undergoes a set of resonances, which occur at the frequencies ω = ω_n+|s|,n, provided that they are found inside the interval where spatial dispersion is strong (see Refs. [9,10] for details),

0 < ω - ω_{s} < | k_{s} | V_{F} = | s | (π V_{F} / c) (δ / d_{b}) ω_{p} .

Here δ = c/ω_p is the electromagnetic skin-depth in bulk metal at optical frequencies.

The electrodynamic boundary conditions at metal-dielectric interfaces x = x_{b_ℓ} and x = x_{a_ℓ+1} lead to linear relations between the fields (1) and (2) at the both interfaces of the ℓth unit (a, b) cell, which can be written in matrix form:

(\begin{matrix} E (x_{a_{ℓ + 1}}) \\ H (x_{a_{ℓ + 1}}) \end{matrix}) = \hat{M} (\begin{matrix} E (x_{a_{ℓ}}) \\ H (x_{a_{ℓ}}) \end{matrix}) .

The elements of the transfer matrix M̂ are expressed in the terms of surface impedances,

M_{11} = \frac{ζ_{0}}{ζ_{d_{b}}} cos (k_{a} d_{a}) - i \frac{ζ_{0}^{2} - ζ_{d_{b}}^{2}}{Z_{a} ζ_{d_{b}}} sin (k_{a} d_{a}),

M_{12} = - \frac{ζ_{0}^{2} - ζ_{d_{b}}^{2}}{ζ_{d_{b}}} cos (k_{a} d_{a}) + i \frac{Z_{a} ζ_{0}}{ζ_{d_{b}}} sin (k_{a} d_{a}),

M_{21} = - \frac{1}{ζ_{d_{b}}} cos (k_{a} d_{a}) + i \frac{ζ_{0}}{Z_{a} ζ_{d_{b}}} sin (k_{a} d_{a}),

M_{22} = \frac{ζ_{0}}{ζ_{d_{b}}} cos (k_{a} d_{a}) - i \frac{Z_{a}}{ζ_{d_{b}}} sin (k_{a} d_{a}),

which eventually define the electrodynamics of metal-dielectric superlattice. For nonmagnetic dielectric layer its impedance Z_a has a standard form,

Z_{a} = 1 / \sqrt{ε_{a}}

. In a metal slab the internal fields (2) are defined by their values on the both boundaries. Accordingly, two surface impedances are introduced – the left-hand ζ₀, and the right-hand ζ_{d_b} [5],

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

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

Both impedances are formed due to contributions of electromagnetic normal modes with different indices s and they exhibit resonant behavior.

Under weak spatial dispersion conditions, the spatial variation scale |k_s |⁻¹ of each electromagnetic s-mode is much greater than the absolute value of the electron effective mean-free path l_ω = V_F/(ν − iω), i.e. |k_sl_ω| = V_F |k_s|/|ω + iν| ≪ 1. Then the transition frequency ω_n+|s|,n becomes negligible in comparison with the frequency ω + iν. In this situation, the quantum nonlocality factor 𝒬(k_s) is suitably represented by its 𝒬(0) value. For a metal film thickness d_b allowing many propagating channels, k_Fd_b/π ≫ 1, which is realized if d_b exceeds the optical skin depth δ = c/ω_p, the quantum nonlocality factor 𝒬(0) → ω/(ω + iν) and the surface impedances turn into their local values,

ζ_{0}^{(loc)} = \frac{i}{\sqrt{ε (0)}} cot (k d_{b} \sqrt{ε (0)}),

ζ_{d_{b}}^{(loc)} = \frac{i / \sqrt{ε (0)}}{sin (k d_{b} \sqrt{ε (0)})} .

Here

ε (0) = 1 - ω_{p}^{2} / ω (ω + i ν)

is the permittivity of classical electron plasma in the Drude-Lorentz model.

For one-dimensional periodicity the dispersion relation 2 cos(κd) = TrM̂ for a photonic eigenmode with Bloch wave number κ takes the form

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

This dispersion relation accounts for quantum Landau damping as far as the resonances in the surface impedances (9) are well manifested [10]. If, however, the surface impedances (9) are replaced by their limiting values (10) obtained in the Drude-Lorentz model, then Eq. (11) is reduced to the well-known Rytov’s equation [16].

For a regular stack composed of N binary unit (a, b) cells with left and right adjacent homogeneous leads, the reflection r and transmission t amplitudes satisfy the relation

(\begin{matrix} t \\ 0 \end{matrix}) = {\hat{M}}^{(T)} (\begin{matrix} 1 \\ r \end{matrix}), {\hat{M}}^{(T)} = {\hat{M}}_{III} {\hat{M}}_{I I} {\hat{M}}_{I} .

The total transfer matrix M̂^(T) is the product of three different matrices corresponding to the regions I, II and III, as shown in Fig. 1,

{\hat{M}}_{I} = (\begin{matrix} 1 & 1 \\ \sqrt{ε_{I}} & - \sqrt{ε_{I}} \end{matrix}), {\hat{M}}_{II} = {\hat{M}}^{N}, {\hat{M}}_{III} = \frac{1}{2} (\begin{matrix} 1 & 1 / \sqrt{ε_{III}} \\ 1 & - 1 / \sqrt{ε_{III}} \end{matrix}) .

The matrix M̂_II is the transfer matrix of the N-bilayer structure. The interface matrix M̂_I describes the wave transfer, through the (I|II) interface from the left lead I with permittivity ε_I into the superlattice II. Another interface matrix M̂_III characterizes the wave transfer through the (II|III) interface from the superlattice II into the right lead III with permittivity ε_III. In what follows, we consider a symmetric case where the leads, both I and III, are the same, i.e., ε_I = ε_III, and, as a consequence, M̂_IM̂_III = 1̂.

Finally, the transmittance T, reflectance R and absorption A are expressed through the elements of the total transfer matrix as follows (see, e.g., Ref. [17])

T = {| t |}^{2} = {| {\hat{M}}_{22}^{(T)} |}^{- 2}, R = {| r |}^{2} = {| {\hat{M}}_{21}^{(T)} / {\hat{M}}_{22}^{(T)} |}^{2}, A = 1 - T - R .

3. Results

If the metal thickness is of the order of the skin-depth or exceeds it, the wave coupling through a metal slab is exponentially weak, resulting in a band structure with wide gaps and rather narrow pass bands. The latter overlap with Fabry-Perot resonances, occurring in the dielectric layers at resonant frequencies [6,7,17]:

ω_{j} = j (c / \sqrt{ε_{a}}) π / d_{a} where k_{a} d_{a} = j π (j = 1, 2, 3 \dots) .

Despite the strong contrast between dielectric Z_a and metallic ζ₀, ζ_{d_b} surface impedances, the photonic dispersion relation (11) has real (propagating) solutions near the “dielectric” Fabry-Perot resonances ω = ω_j.

The photonic band structure near the first dielectric Fabry-Perot resonance (j = 1) for two different vacuum-aluminum arrays, as well as the transmission and absorption spectra, calculated within both quantum-nonlocal and classical-local (Drude-Lorentz) approaches, are presented in Fig. 2. The parameters of the metal layer are: d_b = 1.0 δ with δ ≈ 13 nm, V_F = 2.03 × 10⁸ cm/s, N_F = 70, ω_p = 3.82 × 10³ THz and ν = 1 × 10⁻⁶ ω_p. The chosen thicknesses of the dielectric component, d_a = 447δ and d_a = 448.7δ, correspond to left and right panels, respectively. Such a choice of d_a provides the appearance of the frequency ω_j=1 of the first Fabry-Perot resonance within the interval (6) when the strong spatial dispersion condition is met already for the first normal electromagnetic modes with |s| = 1. For simplicity, we assume the superlattice to be embedded in vacuum, i.e., ε_I = ε_III = ε_a = 1. In panels (a) and (b), the solid red and dotted purple curves give respectively the real and the imaginary parts of the Bloch wave number obtained from the dispersion equation (11) within the quantum approach, while the dashed blue and dotted-dashed orange curves were obtained with the use of the classical-local impedances (10). The black circles on the red curves indicate the frequencies where a second kind of Fabry-Perot resonances occurs. Such type of resonances results from the quantization of the Bloch wave number inside the whole array of N binary unit cells and, therefore, the corresponding resonance condition is given by

Re κ = p π / N d with positive integer p .

The green shaded regions mark the gaps calculated on the basis of quantum-nonlocal approach. The pass bands appear near the resonant frequency ω_j=1, which is marked in all the panels by a dashed magenta vertical line.

Fig. 2 Photonic band structure (panels (a) and (b)), transmission (logT, panels (c) and (d)) and absorption (log A, panels (e) and (f)) spectra near the first dielectric Fabry-Perot resonance, Eq. (15) with j = 1. The curves on the left (right) panels were calculated for d_a = 447δ (d_a = 448.7δ).

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Strong modification of the photonic dispersion law by quantum Landau damping is clearly seen in Fig. 2(a). Here the frequency of quantum electron transition ω_n+|s|,n, depicted by the light-blue vertical line, is located within the classical-local photonic pass band presented by dashed blue curve and originated from the first Fabry-Perot resonance, Eq. (15) with j = 1. Avoided crossing of the quantum Landau resonance ω = ω_n+|s|,n with the photonic spectrum of Reκ(ω) results in the appearance of the quantum-nonlocal solid red line, or, the same, in the splitting of the classical-local photonic band into two minibands separated by a narrow gap where the dispersion of Reκ(ω) is anomalous and the value of Imκ(ω) is relatively large. The real part of the Bloch wave number Reκ does not vanish within the new gap due to the finite value of the electron relaxation rate ν. For the chosen parameters, the quantum transition frequency ω_n+|s|,n corresponds to the electron transition between conducting channels with n = 23 and n + |s| = 24.

The panel (c) of Fig. 2 shows the transmission spectra logT(ω) for a regular stack composed of N = 10 unit cells within the same interval of frequencies as in the panel (a). The dashed pink and solid blue curves represent, respectively, the optical spectra obtained via the Drude-Lorentz model and the quantum-nonlocal approach. According to classical-local calculations, the number of transmission peaks, associated with the second-type Fabry-Perot resonances (16) is equal to N − 1 = 9 with their amplitudes close to one. It was shown in Ref. [7] that the Landau damping calculated in the classical-nonlocal regime is responsible for the decrease and broadening of the transmission resonant peaks. However, as can be seen from the panel (c), the quantum-nonlocal approach predicts not only the diminishing of the peaks, but also a narrow frequency range of extremely suppressed transmission where the new gap appears. Analysis of the absorption spectrum in the panel (e) demonstrates that the absorption is enhanced in the region of the new gap. This means that the interplay between the dielectric Fabry-Perot resonance (15) and the quantum electron transition between the states with n = 23 and n + 1 = 24 not only modifies the photonic dispersion, converting normal dispersion to anomalous one, but also increases the level of absorption. Note that each transmission peak of the quantum-nonlocal spectrum is shifted giving rise to additional picks, as compared to the classical-local spectrum.

If the interval between the resonances ω = ω_n+|s|,n and ω = ω_j becomes shorter, the strength of level repulsion increases leading to even stronger modifications in the classical-local dispersion, transmission and absorption. Very slight increase of the width d_a of the dielectric layer from d_a = 447δ to d_a = 448.7δ significantly shifts the dielectric Fabry-Perot resonance ω = ω_j towards quantum Landau resonance ω = ω_n+|s|,n. The photonic band structure corresponding to this extreme case of very close resonances is shown in panel (b). Strong repulsion of levels gives rise to a very narrow pass band centered near ω_j=1. Thus, in the proposed quantum approach the superlattice exhibits relative transparency in the region where it is completely opaque according to the classical-local approximation. A transmission peak near the frequency ω_j=1 is clearly seen in the panel (d).

There are two reasons for wave decay in a periodic system: Bragg reflection and energy dissipation. Since the photonic dispersion relation (11) contains dissipative mechanisms, which are electron collisions with frequency ν and quantum nonlocality, the decay given by Imκ takes into account not only the Bragg reflection but also irreversible energy loss due to collisional dissipation and reversible loss due to collisonless Landau damping. Unlike this, the absorption A accounts only dissipative losses, both reversible and irreversible. Different behavior of Imκ and log A is clearly seen in panels (b) and (f). Within the gap region (normalized frequency between 6.98 and 7.00) the wave decay is due to Bragg reflection since the value of Imκ is high, but log A is relatively low. Strong increase of Imκ is associated, according to Kramers-Kronig relations, with anomalous dispersion of the wave in this region. In the region of narrow pass band (normalized frequency between 7.00 and 7.05) the decay is only due to dissipation. Here Imκ practically vanishes (it remains finite due to finite Landau damping near resonance) but the absorption increases as a result of Joule dissipation in metal generated by propagating wave. Because of this enhanced absorption the transmission within the pass miniband remains relatively low (see the panel (d)).

The first band in the quantum transmission spectrum shown in the panel (d), is narrower than that in the Drude-Lorentz model. It still contains N − 1 = 9 resonances, while in the second band (between 7.00 and 7.05) all the peaks are overlapped. Since nonlocal effects are stronger close to ω_j=1 frequency, the transmission (absorption) resonances decrease (increase) at the right edge of the first band, as shown in panels (d) and (f).

The case when the above-commented electron transition frequency ω_n+|s|,n with n = 23 and n +1 = 24 is found within the second Drude-Lorentz pass band is illustrated by Fig. 3. To this end, we have chosen the dielectric layer thickness so that the frequency ω_n+|s|,n turns out to be close to the second dielectric Fabry-Perot resonance, Eq. (15) with j = 2. The other parameters remained unchanged. Here, as in the previous case, the photonic spectra are dramatically modified by the phenomenon of quantum Landau damping. In particular, a new kind of band gap (quantum band gap) with anomalous dispersion of Reκ emerges. Moreover, when the second dielectric Fabry-Perot resonance ω = ω_j=2 and the Landau damping resonance ω = ω_n+|s|,n approach each other, an additional quantum pass band arises.

Fig. 3 Photonic band structure (panels (a) and (b)), transmission (logT, panels (c) and (d)) and absorption (log A, panels (e) and (f)) spectra near the second dielectric Fabry-Perot resonance, Eq. (15) with j = 2. The curves on the left (right) panels correspond to d_a = 895.8δ (d_a = 897.5δ). The other parameters are the same as in Fig. 2

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In the photonic band structures of lossless periodic systems, the regions of frequencies where Imκ = 0 are the pass bands and the regions where Imκ > 0 and Reκ = const are the band gaps. Within the band gaps, originated from the Bragg reflection, the density of photonic states (DOS) vanishes. The presence of finite dissipation (finite scattering frequency ν) does not change the classification of zones but smoothens the boundaries between pass bands and gaps [18]. As shown above, the quantum collisionless Landau damping may strongly modify classical-local band structure in the vicinity of quantum resonances. New narrow band gaps and pass bands may appear in the spectrum if a Fabry-Perot resonance and a quantum resonance are getting sufficiently close. Within the new “quantum” gap the value of Reκ does not remain constant, exhibiting anomalous dispersion, see panels (a) and (b) of Figs. 2 and 3. By Kramers-Kronig relations the anomalous dispersion of Reκ means enhanced value of Imκ, i.e., strong dissipation. It is important to stress that in this particular case the dissipation is collisionless by nature, i.e., it remains finite even in the limit ν → 0. While the collisionless dissipation mechanism (Landau damping) is thermodynamically reversible it leads to exponential decay of wave, which means no propagation through an infinite sample. This scenario explains the presence of narrow quantum gaps in a dielectric-metal superlattice in the collissionless limit. Thus, the band gaps in a periodic system may appear not only due to conventional Bragg reflection but also due to level repulsion, if one of the levels is the quantum Landau damping resonance.

To illustrate the different nature of conventional Bragg-reflection gaps and quantum Landau-damping gaps we plot in Fig. 4 the DOS corresponding to the photonic dispersions shown in panels (a) and (b) of Figs. 2 and 3. Within the conventional gaps (wide green zones on both sides of the spectra in Fig. 4) the DOS vanishes. Nevertheless, inside the narrow central gap, originated from Landau damping, the DOS takes quite large values for all spectra independently of the parity of the Fabry-Perot resonances, i.e. for j = 1 as well as for j = 2. This means that the Bloch wave does not suffer from destructive interference leading to Bragg reflection. However, here it strongly interacts with quantized electrons which resonantly absorb electromagnetic energy.

Fig. 4 Density of photonic states calculated from the dispersion relation (11). Note that the density of states does not vanish within the narrow quantum gaps at the center of the spectrum.

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The quantum-nonlocal and Drude-Lorentz absorption spectra (log A), calculated for different electron relaxation frequencies ν, are compared in the respective panels (a) and (b) of Fig. 5. As one can see, the absorption peaks, associated with the quantization of the Bloch wave number, namely the second-type Fabry-Perot resonances (16), are clearly pronounced within a quite broad range of the electron scattering rate: ν/ω_p ∼ 10⁻⁷ − 10⁻⁴. What is more remarkably, as follows from panel (a), the new gap where Landau damping strongly modifies the quantum-nonlocal photonic spectrum, is clearly manifested. In the classical-local regime the higher rate ν of electron relaxation leads to higher electromagnetic absorption. However, in the quantum-nonlocal regime the electromagnetic energy is absorbed due to resonant interaction of a small group of effective electrons with the wave. The time of resonant interaction of these electrons with the wave increases if ν → 0 that leads to higher absorption in the collisionless limit, as it follows from the depicted plots.

Fig. 5 Absorption (log A) spectra of a vacuum-aluminum regular stack for three different values of the electron relaxation rate ν marked in the figure. The other parameters are the same as in the left panels of Fig. 2.

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In the end of this section let us briefly discuss the numerical results regarding electric field distribution inside the dielectric-metal stack. We have found out that for the frequencies within the conventional Bragg-reflection gaps the field distribution obtained both in Drude-Lorentz model and in the quantum approach has the same profile with very small differences in amplitude. Then, in the conventional pass band, arising in the Drude-Lorentz model, the quantum Landau damping expectedly suppresses the electromagnetic field by 2–3 orders of magnitude and changes the field profile. The most striking effect of the quantum Landau damping occurs within the new quantum gaps in the region of frequencies where a second-type Fabry-Perot resonance of Drude-Lorentz model provides almost perfect transmission. Fig. 6 displays the electric field profile in superlattices at the frequencies of the second-type Fabry-Perot resonances with p = 6 in Eq. (16), which are clearly seen in the left panels of Figs. 2 and 3. The green vertical lines schematically indicate the 10 ultra-thin metal films, whereas the dielectric layers occupy the space between them. Due to the resonances, the electric field in the Drude-Lorentz model has quite sophisticated oscillating pattern with large amplitudes. On the other hand, the electric field, predicted within the quantum approach (see the insets), is much weaker. It decays rapidly over the distance of a few unit cells with different oscillatory behavior.

Fig. 6 Electric field distribution normalized to the amplitude of the incident wave.

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4. Conclusion

In this work we have developed a theory of electromagnetic transmission through dielectric-metal superlattices with ultra-thin metal films, in which electron motion in the direction perpendicular to the metal surfaces is quantized. Dispersion relation for Bloch photonic modes and spectra of electromagnetic transmission and absorption are calculated with the use of the classical-local Drude-Lorentz model and quantum-nonlocal Kubo formalism. The results obtained within these two approaches have some common features. However, there is also an essential difference in the regions where the frequency of an electron quantum transition, causing the quantum Landau damping, turns out to be close to the electromagnetic Fabry-Perot resonance associated with the dielectric layers. Avoided crossing of the quantum Landau resonance with the Fabry-Perot photonic pass band strongly modifies the photonic dispersion relation. The modified spectrum may contain either new quantum gaps and/or new pass bands, depending on the relative position of the quantum Landau-damping resonant frequency and the frequency of the Fabry-Perot photonic resonance. The mechanism of formation of the new gaps is qualitatively different from the conventional Bragg reflection by a set of periodic scatterers. The mechanism which prevents wave propagation in the new gaps is the collisionless absorption of electromagnetic energy due to transitions between size-quantized electron states (quantum Landau damping). In contrast to the Bragg-reflection mechanism, the new one does not require zero density of photonic states inside the gaps. The photonic modes existing within the new quantum gaps exhibit anomalous dispersion. Such “dark” modes may serve as short-range carriers with strongly anomalous dispersion. While the dark modes are non-observable in an infinite sample, in a finite-size sample their presence can be detected due to exponentially decaying tails. Recently, the contribution from exponentially decaying plasmonic modes to the local density of states was measured in the near field [19]. The same technique can also be used for detecting the dark photonic modes.

Finally, let us briefly discuss the possibility of experimental observation of the predicted phenomena. In real artificial periodic systems, the typical deviations from those with perfect structure are interface roughness, polycrystallinity of metallic films, fluctuations of the superlattice period, impurities, etc. All these imperfections in many cases serve as additional sources of random scattering of conduction electrons, and, consequently, can be taken into account by the phenomenological parameter ν, playing the role of the effective electron relaxation frequency. As was demonstrated in the previous section (Fig. 5), the effects originated from the Landau damping quantization should be discernible within a quite broad range of the electron scattering rate ν. Besides, weak fluctuations of the superlattice period give rise to small shifts and smoothing of the edges between passing bands and gaps. This type of defects may reduce the sharpness of the Fabry-Perot resonances. Our numerical estimations show that relative fluctuations of the period should not exceed a few percent.

The important condition for experimental realization of the Landau damping quantum effects is the possibility to form homogeneous metallic ultra-thin films, avoiding island structure. The quality of the films to a large extend depends on the employed growing technique (e.g., thermal evaporation [20,21], magnetron sputtering [20,22], molecular beam epitaxy [23–25]). The latter allows to grow metal films that become rather homogeneous starting from 10 nm thickness with surface roughness reduced to the atomic scale. For this reason, we believe that epitaxial metal films may serve as a proper components to fabricate experimental binary multilayer stack.

Funding

Benemérita Universidad Autónoma de Puebla (BUAP) (100312733-VIEP2018); National Science Foundation (NSF) (1741677).

Acknowledgments

This work was partially supported by the VIEP-BUAP (México) under the grant No. 100312733-VIEP2018. AAK is thankful for hospitality at the Instituto de Física Universidad Autónoma de Puebla where part of this work was accomplished. He also acknowledges support from the NSF through the EFRI grant no. 1741677.

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Electrodynamics of superlattices with ultra-thin metal layers: quantum Landau damping and band gaps with nonzero density of states

Abstract

1. Introduction

2. Formulation of the problem

3. Results

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (6)

Equations (24)

Optical Materials Express