Plasmon resonance at the interface dielectric - nanocomposite material with superconducting inclusions

This article theoretically considers surface plasmon resonance in composite structures. The features of the surface plasmon resonance arising at the interface with media containing nanoparticles from a high-temperature superconductor are investigated. The dielectric constant of spherical superconducting inclusions is considered taking into account Gorter-Casimir two-fluid model. The temperature dependence of the electrodynamic parameters of the superconductor is taken into account. The two-fluid model, the dependence of the concentration of non-superconducting electrons in a superconductor is often used as the fourth power of temperature nn~T4. In this work, a phenomenological model is used, according to which the electron concentration of non-superconducting electrons in a superconductor is determined by the formula nn~Tγ with γ=1.3÷2. This model is in good agreement with experimental data for high-temperature ceramic superconductors. The dispersion characteristics of surface plasmons arising in a planar structure with a thin nanocomposite superconductor layer are investigated. It is shown that the dispersion characteristics depend significantly on temperature.


Introduction
Plasmonics is one of the most modern scientific directions with great potential for practical use [1]. Materials in which surface plasmon resonance can be observed are increasingly used to create various optoelectronic devices [2]. Surface plasmon resonance occurs at the interface between two media with positive and negative dielectric permittivity [3]. A special place among plasmon materials is occupied by media with inclusions of nanoparticles of various shapes [4]. One of the important features of nanoplasmonic materials is the dependence of the resonance frequency on the size of nanoparticles, their concentration, and other parameters [5]. Surface plasmons can arise at various interfaces: metaldielectric [6], metalsemiconductor [7], semiconductordielectric [8], metamaterialdielectric [9], superconductordielectric [10]. Surface plasmons arising at the interface with a superconductor have a number of distinctive features. First of all, this is a small attenuation in superconductors at low temperatures. Another important feature is the strong temperature dependence of the properties of the superconductor. This dependence is especially pronounced at temperatures close to the critical temperature [11]. Plasmonic devices based on high-temperature superconductors (HTSC) are used to create supersensitive photon detectors [12], sensors [13], waveguides [14], amplifiers [15]. Most hightemperature superconductors are complex ceramic compounds with an inhomogeneous composition. Therefore, the fabrication of HTSC nanoparticles with a radius of less than 10 nm is a complicated process. However, according to the latest data, HTSC nanoparticles have a higher critical temperature,  [16]. This article investigates the properties of surface plasmons arising at the plane boundary of a dielectric and a composite medium containing spherical HTSC nanoparticles.

Methods
Let us consider the propagation of surface plasmons in a structure that contains a thin layer of a superconducting nanocomposite located between two nonmagnetic dielectrics (   The electrodynamic properties of superconductors are described by a two-fluid model. In this case, electrons in a superconductor are divided into two types. The first type is superconducting electrons that move without dissipation. The second type is ordinary non-superconducting electrons, which have losses. Each type of electron moves independently of the other. The dielectric permittivity of superconductor in accordance with the two-fluid model is described by the following formula [17] ( ) ( ) For conventional low-temperature superconductors, good agreement with experimental data gives the value =4 [17]. However, as shown in [16], the situation is more complicated for high-temperature superconductors. The value of  in essentially depends on the type of the HTSC. So, according to [1],  Within the framework of phenomenological model [17], the conductivity of normal carriers depends on temperature as follows   Surface plasmon resonance is observed at the interface between two media with positive and negative dielectric constant [18]. Figure 3 shows the results of calculating the frequency dependence of the real part of the effective dielectric constant of a nanocomposite with superconducting inclusions. The  Figure 3 shows that the real part of the effective dielectric constant of the composite becomes negative at frequencies exceeding the surface plasmon resonance frequency for the composite. As can be seen from figure 3, this frequency increases with increasing temperature.
The temperature dependence of eff  for different values of the volume fraction of inclusions f is shown in figure 4. This calculation was performed for the same parameters as indicated for figure 3. We see from figure 4 that the temperature dependence has a resonant character, and can take negative values at low temperatures. Thus, surface plasmons can exist in the structure under consideration.
Dispersion relations for surface plasmons at the interfaces in the structure shown in figure 1 were obtained in [18]. In the case of equality of the parameters of the dielectrics

Discussion
The dispersion characteristics of surface plasmons that can propagate at the interface in the structure composite with HTSC inclusions -dielectric are shown in figure 5. low-frequency branch has no cutoff and starts at frequency = 0. The high frequency branch has a temperature dependent cutoff frequency.
As expected, the dispersion characteristics show a temperature dependence. As the reduced temperature t rises from 0.6 to 0.99, the cutoff frequency increases from 1.610 15 s -1 to 2.310 15 s -1 . Thus, by changing the temperature, it is possible to control the appearance of surface plasmon resonance. The temperature dependence of the cutoff frequency for high-frequency plasmons can be used to control the number of modes propagated in planar structures. Also, structures with superconducting inclusions can be used to create sensitive sensors with the possibility of temperature change in their parameters.

Conclusion
The study carried out in this work has shown that composite media with HTSC inclusions have good prospects for practical application. The characteristics of surface plasmons arising at the interface with such media depend both on the concentration of superconducting inclusions and on temperature. This circumstance makes it possible to control the parameters of surface plasmons.