Plasma excitations in two-dimensional systems have been studied for more than five decades [1–15]. These studies result in the discovery of numerous new fundamental physical effects. In particular, a transverse electromagnetic plasma mode was experimentally detected in a hybrid system consisting of a two-dimensional electron system on a dielectric substrate [16–20]. Two fundamentally different types of plasma waves—longitudinal electrostatic and transverse electromagnetic waves—exist in two-dimensional systems, as in three-dimensional systems [17]. The electric field vector for longitudinal plasmons is directed along the propagation direction of the waves. On the contrary, the alternating electric field for transverse electromagnetic plasma waves is perpendicular to the propagation direction of the waves.

A transverse electromagnetic mode has been recently studied in detail theoretically [17, 18] and detected experimentally [19, 20]. The electrodynamic response of a system in the experimentally typical case of a uniform plane wave with a linear polarization normally incident on the surface of the infinite two-dimensional electron system on the substrate of the thickness d is specified by the formula [16]

$$\varepsilon (\omega ) = \varepsilon {\kern 1pt} \left( {1 - \frac{{\omega _{0}^{2}}}{{{{\omega }^{2}}}}} \right),\quad {{\omega }_{0}} = \sqrt {\frac{{{{n}_{s}}{{e}^{2}}}}{{m{\text{*}}{{\varepsilon }_{0}}\varepsilon {\kern 1pt} d}}} .$$
(1)

Here, ns, e, and m* are the density, charge, and effective mass of charge carriers in the two-dimensional electron system, respectively, and d and ε are the thickness and relative permittivity of the substrate, respectively. It is noteworthy that this formula is valid only if the wavelength of radiation is much larger than the thickness of the substrate, which is equivalent to the condition \(\omega \ll \pi c{\text{/}}(\sqrt \varepsilon d\)). The transmission spectrum of electromagnetic radiation for this system at ε(ω) = 1 demonstrates a resonance corresponding to the transverse electromagnetic plasmon at the frequency [18–20]

$${{\omega }_{{{\text{2D}}}}} = \sqrt {\frac{{{{n}_{s}}{{e}^{2}}}}{{m{\text{*}}{{\varepsilon }_{0}}{\kern 1pt} (\varepsilon - 1){\kern 1pt} d}}} .$$
(2)

One of the conditions for the observation of two-dimensional plasmons with the frequency ωp is the inequality ωpτ \( \gg \) 1, where τ is the electron relaxation time. Because of this condition, two-dimensional plasmons in the frequency range up to 500 GHz in modern semiconductor and layered materials can be observed only at cryogenic temperatures. This circumstance significantly complicates the application of two-dimensional plasmonics.

It was shown that the three-dimensional grating of metallic wires at room temperature has electrodynamic properties in the microwave band that are equivalent to the three-dimensional plasma [21–26]. Similarly, for a two-dimensional metallic grid with a period of a and a strip width of w, the effective two-dimensional electron density can also be introduced as neff = nsw/a, where ns is the two-dimensional density of electrons moving in the skin layer of the metallic film. The interaction with the external electromagnetic field is manifested in the self-induction of the “grating” structure, which gives the effective electron mass [23]

$${{m}_{{{\text{eff}}}}} = \frac{{{{\mu }_{0}}w{{e}^{2}}{{n}_{{\text{s}}}}}}{{2\pi }}{\kern 1pt} \ln \frac{a}{w}.$$
(3)

Considering the two-dimensional grating on the dielectric substrate as an effective electron system, we can use Eq. (2) to calculate the frequency of the transverse plasma resonance:

$$\omega _{{\text{p}}}^{{\text{2}}} = \frac{{2\pi {{c}^{2}}}}{{ad{\kern 1pt} (\varepsilon - 1)\ln (a{\text{/}}w)}},\quad {{\omega }_{{\text{p}}}} \ll \frac{{\pi c}}{{\sqrt \varepsilon d}}.$$
(4)

It is remarkable that this plasma frequency of the metasurface depends on the geometric parameters of the structure and is independent of the electron density in the metal.

In this work, we demonstrate that the two-dimensional metallic grating on the silicon substrate serves as the metasurface, where the response of transmission and reflection is similar to the response of the effective two-dimensional electron system associated with the excitation of the transverse electromagnetic plasma mode.

The studied samples were high-resistivity (>30 kΩ cm) silicon plane–parallel plates with the metallic grid lithographically printed on one side. We carried out two series of experiments with 1 × 1-cm samples with thicknesses d = 103 and 213 μm. A grating metallic structure with a square unit cell with the period a = 100, 150, 200, and 300 μm and the strip width w = 5, 10, 20, and 30 μm was fabricated on their surface by means of optical photolithography (see Fig. 1). The ratio w/a for all samples was no more than 0.1. On the surface of the silicon substrate, the 25‑nm‑thick Cr and 700-nm-thick Au layers were thermally deposited in a vacuum chamber. The parameters of the samples and their enumeration are described in detail in the supplementary material. The measurements were performed at the terahertz spectroscopy setup “Epsilon” [27] with a backward-wave oscillator for the frequency range of 50–500 GHz used as a continuous terahertz radiation source. The power of radiation transmitted through the sample was measured by a pyroelectric detector. To suppress noise, the electromagnetic radiation was modulated by an optomechanical modulator with a frequency of 23 Hz.

Fig. 1.
figure 1

(Color online) (Top panel) Schematic of the surface of sample \(4\) with the metallic grating with the parameters a = 300 μm and w = 30 μm. The diagonal photograph of the studied sample is shown on the right. (Bottom panel) Measured transmission spectra for the first four samples with the thickness of the silicon substrate d = 103 μm and the strip width w = 10 μm. The spectra are shifted in the vertical direction for clarity. The solid lines are the results of the numerical calculation. The dashed lines are the transmission spectra of the silicon substrate without the grid according to Eq. (5). The arrows mark the positions of the plasma resonance in the transmittance.

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Figure 1 shows the transmission spectra measured for the first four samples with grating periods from 100 to 300 μm. A resonance in the transmission marked by the arrow appears in the low-frequency part of the spectrum. The solid lines are the results of the numerical simulation performed for each sample with the HFSS simulation software. It is seen that these results are in good agreement with the experimental data. Thus, the electromagnetic response of the metallic grid on the dielectric substrate is similar to the response of a high-quality two-dimensional electron system. This can be qualitatively explained by the fact that metallic strips directed along the polarization of the incident radiation imitate the kinetic inductivity of the two-dimensional electron system. Strips perpendicular to the incident polarization hardly affect the transmission spectrum. The detailed analysis of this property, as well as the comparison of measured transmission spectra with spectra of real two-dimensional electron systems, is presented in the supplementary material.

The appearance of additional resonances in the transmission spectrum of the structure with the grating period a = 300 μm is remarkable. They correspond to the diffraction of the incident electromagnetic wave on the metallic grid. Diffraction effects for samples with smaller periods are manifested at frequencies higher than 500 GHz.

To verify Eq. (4), we plot the dependence of the frequency of observed resonances on the inverse period of the grating metasurface 1/a (see Fig. 2). We note that the frequency positions of resonances for gratings with the periods a = 300 and 200 μm are well reproduced by Eq. (4) (see red line in Fig. 2). However, the deviation of the resonance frequency from Eq. (4) increases with a decrease in the period of the metallic grid. This is due to the violation of the condition \({{\omega }_{p}} \ll \pi c{\text{/}}(\sqrt \varepsilon d)\). The blue horizontal straight line in Fig. 2 corresponds to the frequency \(\pi c{\text{/}}(2\sqrt \varepsilon d)\) equal to half the frequency of the first Fabry–Pérot resonance of the dielectric substrate. The observed deviation of the resonance from the calculated frequency toward the substrate resonance is due to the hybridization of the plasmon with substrate modes. Hybridization effects are studied in more detail below.

Fig. 2.
figure 2

(Color online) Frequency of the resonance measured for samples 1–4 versus the inverse period of the grating metasurface 1/a. The red line is the theoretical dependence of the plasma frequency according to Eq. (4). The blue horizontal straight line is the frequency \(\pi c{\text{/}}(2\sqrt \varepsilon d)\) equal to half the frequency of the first Fabry–Pérot resonance of the dielectric substrate.

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Returning to the experimental verification of Eq. (4), we note that an important feature of this formula is the logarithmic factor in the denominator. To verify this feature, we conducted additional experiments with the samples satisfying the conditions \({{\omega }_{{\text{p}}}}\, \ll \,\pi c{\text{/}}(\sqrt \varepsilon d)\). Grids on samples 4, 5, and 6 have the same period a = 300 μm but different strip widths w. The dependence of the frequency of the plasma resonance on the width w is plotted in Fig. 3. It is seen that the experimental data are well reproduced by theoretical formula (4) (see solid lines in Fig. 3).

Fig. 3.
figure 3

(Color online) (Circles) Frequency of the plasma resonance measured for samples 4–6 versus the strip width w in comparison with the theoretical solid line plotted according to Eq. (4). The inset shows the same data in the form of the plot of \({{f}^{{ - 2}}}\) versus ln(a/w).

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We now discuss the hybridization of the plasma resonance with the resonances of the dielectric substrate. The transmission spectra in the silicon plate with the thickness d without the metallic grid for each of the samples are shown by dashed lines in Fig. 1. They are analytically described by the Fabry–Pérot function

$${{T}_{{{\text{FP}}}}} = \frac{1}{{1 + \frac{1}{4}{{{\left( {\sqrt \varepsilon - \frac{1}{{\sqrt \varepsilon }}} \right)}}^{2}}{{{\sin }}^{2}}\left( {\frac{{\omega d\sqrt \varepsilon }}{c}} \right)}}.$$
(5)

The Fabry–Pérot function for silicon with ε = 11.9 provides a set of equidistant peaks at the frequencies \({{\omega }_{N}} = N \times \pi c{\text{/}}(\sqrt \varepsilon d)\), where \(N = 0,1,2, \ldots \). It is seen in Fig. 1 that the addition of the metallic grid on the substrate significantly shifts the zeroth (N = 0) Fabry–Pérot peak from zero frequency. According to the proposed model, this shift is a plasma effect. It is seen that, when the condition (4) is violated, the resonance is shifted toward the frequency ω1/2, where ω1 is the frequency of the first Fabry–Pérot resonance on silicon.

To completely understand the hybridization mechanism, it is reasonable to consider the strong hybridization regime at ωp \( \gg \) ω1/2. For this, we carried out additional experiments with samples 7 and 8 having the substrate thickness d = 213 μm, where the ratio ωp/ω1/2 is larger. The results of all experiments are plotted in Fig. 4 in the form of the dependence of the ratio ω/ω1 on ωp/ω1, where ω is the resonance frequency, ω1 is the frequency of the first Fabry–Pérot resonance, and ωp is the plasma frequency given by Eq. (4). Green circles are the data for samples 1–3, blue circles are the results for samples 4–6, and red circles are the data for samples 7 and 8. It is seen that the experimental data are consistent with each other and are described by a common dependence. In the plasma limit, where ωp/ω1 → 0, the position of the plasma resonance of the grating metasurface is well described by Eq. (4). In the strong delay limit, where ωp/ω1 → ∞, the frequency of the plasma resonance asymptotically approaches the frequency ω = ω1/2 of the photon mode of the Fabry–Pérot resonator.

Fig. 4.
figure 4

(Color online) Ratio ω/ω1 versus ωp/ω1, where ω is the resonance frequency, ω1 is the frequency of the first Fabry–Pérot resonance, and ωp is the plasma frequency given by Eq. (4) according to (circles) measurements on samples with different geometries of the metallic grating, (red dashed line) theoretical formula (6), (red solid line) experimentally established formula (7), (inclined black line) theoretical formula (4), and (horizontal straight line) ω/ω1 = 1/2.

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The hybridization of the plasma resonance with light modes is conveniently described by the delay parameter A [28, 29]. This parameter for electromagnetic plasma modes, which are hybridized with modes of the Fabry–Pérot substrate, is defined as A = 2ωp/ω1, where ω1 is the frequency of the first (N = 1) Fabry–Pérot resonance. In this case, the frequency of the hybrid plasma mode in the two-dimensional electron system on the substrate is given by the formula [18, 19]

$$\omega = \frac{{{{\omega }_{{\text{p}}}}}}{{\sqrt {1 + {{A}^{2}}} }}.$$
(6)

This dependence is shown by the red dashed line in Fig. 4. It is seen that it is inconsistent with experimental data likely because the grating metasurface is strongly different in physical properties from the two-dimensional electron system. However, the experimental data are well described by the empirical formula

$$\omega = \frac{{{{\omega }_{{\text{p}}}}}}{{\sqrt[4]{{1 + {{A}^{4}}}}}}.$$
(7)

This dependence is shown by the red solid line in Fig. 4.

To summarize, the transmission of electromagnetic radiation through a silicon substrate with a square metallic grid deposited on its surface has been studied experimentally. It has been established that the electrodynamic response of this structure is similar to the response of the effective two-dimensional electron system on the substrate, which is equivalent to the excitation of a transverse plasma mode in the studied structure. It has been found that the effective plasma frequency of this mode is determined by the geometric parameters of the grating, the thickness of the substrate and its relative permittivity. A theoretical model has been developed to qualitatively describe the experimental results obtained. The results can be useful to develop novel terahertz devices.