Interplay between collective modes
in hybrid electron gas–superconductor structures

Mesoscopic systems in normal (not superconducting) state lodge elementary excitations, such as excitons and plasmons. They determine the response of the material to external perturbations. Plasmons represent collective modes of particle density fluctuations and play an essential role in the material response to external alternating electromagnetic (EM) fields. Their dispersion law and damping strongly depend on the dimensionality of the system. Indeed, in the case of three-dimensional (3D) electron plasma, the plasmon branch is with a good accuracy dispersionless, whereas two-dimensional (2D) plasmons are characterized by a square-root dispersion relation and vanish in long-wavelength limit (at least, in the framework of the quasistatic approximation). Many-component 2D systems like an electron-hole 2D gas are characterized by two plasmon branches with linear (acoustic plasmon) and square-root (optical plasmon) dispersions.

If with a decrease of temperature the system undertakes a phase transition to another state like a Bose-Einstein or superconducting (SC) condensate, the ground state changes and new elementary excitations describing low-energy properties of the system arise. This can lead to drastic modification of the material response to external EM fields, which becomes especially important in view of resent progress in plasmonics, optoelectronics, and high-temperature condensation phenomena.

In the pioneering work of Bardeen Cooper and Schriffer (BCS) BCS , only single-particle excitations characterized by the SC gap were studied. Later, Anderson Anderson , Bogoliubov Bogoliubov and Vaks, Galitskii, and Larkin VaksGalitskiiLarkin developed a more extensive theory, which includes the collective excitations of Cooper pairs. They showed, that in a neutral Fermi system with an attractive interaction between fermions, collective modes possess a soundlike dispersion. A charged Fermi system, instead, demonstrates the collective modes with the frequency pushed towards the plasmon frequency of the 3D charged electron gas. This frequency is usually much larger than the BSC gap and consequently it does not play role in conventional phenomena in SCs.

These modes correspond to oscillations of the SC order parameter. More precisely, the Anderson-Bogoliubov mode represents the oscillation of the phase of the order parameter, whereas the Higgs mode corresponds to the oscillations of its amplitude. These collective modes play crucial role in the gauge-invariant response of superconductors to external EM perturbations Arseev .

Later on, Carlson and Goldman CG1 ; CG2 ; CG3 experimentally discovered another type of collective modes, which occur in the vicinity of the transition temperature $T_{c}-T\ll T_{c}$ in superconductors. This experimental finding stimulated theoretical studies. In the dirty-sample case, there was suggested the model of Schmid and Schön SS , whereas in the clean (low-disorder) limit, there was developed the model by Artemenko and Volkov (AV) AV1 ; AV2 . Qualitatively, AV mode corresponds to the out-of-phase oscillations of normal and SC currents in such a way that it prevents the emergence of net charges in the system. The frequency of this mode is smaller than the plasmon frequency of normal 3D electrons, thus suppressing the interaction between them.

In the meanwhile, the coupling between single-particle and collective modes in hybrid systems consisting of a 2D electron layer and a 2D condensate of Bose particles represent actively developing areas of research recently RefAlt ; Imamoglu ; KovChap . In particular, there emerge such phenomena as the magnetoplasmon Fano resonances in uniform magnetic fields BKS and amplification of the incident light OurPRBKus ; OurPRLKus . The progress in theoretical proposals and the lack of well-established experimental platforms motivates the study of the interplay of different collective modes in hybrid systems. In this article, we investigate the hybridization of collective plasmon and AV modes in a hybrid semiconductor-superconductor structure (where one of the subsystems is in the SC state). As we have already pointed out, the plasmon mode has a gapless dispersion in 2D case. We will show that it opens a possibility of interaction between AV modes and plasmons.

At first sight, despite the gaplessness of the plasmon mode, the interaction between them is not possible since the AV mode exists in the domain $\omega\tau\ll 1$ (here $\tau$ is electron-impurity scattering time), whereas the plasmon mode exist at $\omega\tau\gg 1$, and these two conditions do not overlap. However, their interaction can become possible if the normal electron gas layer (having the plasmon mode) and the SC layer (hosting the AV mode) are separated in space and electrons in each layer have different scattering times. Thus, if we denote as $\tau_{n}$ the scattering time of electrons in the normal layer and $\tau_{s}$ the one for quasiparticles in the SC layer, the simultaneous existence of AV and plasmon modes is possible when $\tau_{s}^{-1}>\omega>\tau_{n}^{-1}$. This is a necessary criterion for the coexistence of both the modes in one sample. However, it is not a sufficient condition for their hybridization. What are the other conditions? This is the main question, which we address in this article.

Let us consider a 2D layer of a semiconductor containing normal electrons with the equilibrium density $N_{n}$ in the vicinity of a 2D SC layer containing both normal single-electron excitations above the SC gap $\Delta(T)$, where $T$ is temperature (we will use $\Delta/T\ll 1$), and SC Cooper pairs described by the order parameter (Fig. 1). The total electron density in SC layer is $N_{s}$, whereas the the density of SC electrons we denote as $n_{s}$.

Furthermore, we expose the system to an external EM field, which represents a plane wave with 3D wave vector $\textbf{Q}=(\textbf{k},Q_{z})$, where the 2D wave vector $\textbf{k}=(Q_{x},Q_{y})$ is its in-plane projection. The field creates electric currents in both the layers,

where $\textbf{E}_{0}$ is the in-plane component of external EM field, which we assume to be equal in both layers (valid as long as $Q_{z}d\ll 1$); $\textbf{E}_{n,s}^{i}$ are the internal EM fields, induced in the corresponding layers due to the oscillations of the particle densities;

is a conventional dynamical Drude conductivity of the degenerate electron gas in the semiconductor, with $m_{n}$ being the electron effective mass; $\sigma_{s}$ describes the current response of single-particle excitations in the SC layer. It should be noted, that in Eq. (1) we omitted the factor $\exp(i\textbf{kr}-i\omega t)$ since all the time and in-plane position-dependent quantities are proportional to this factor.

The induced fields depend on $z$, $\textbf{E}_{n,s}^{i}\equiv\textbf{E}^{i}(z=\pm d)$, where $\textbf{E}^{i}$ can be expressed via the scalar potential in the framework of quasistatic approximation, $\textbf{E}^{i}=-\nabla\varphi^{i}$. The scalar potential, in turn, satisfies the Poisson equation,

where $\delta\rho_{n,s}$ are deviations of charge densities from their equilibrium values, and $\epsilon$ is a dielectric function of the material. Combining (1) and (3) with the continuity equation, we find

and

where we have introduced the longitudinal dielectric function,

Here and in what follows we assume that k and $\textbf{E}_{0}$ only have $x$ in-plane components.

Eq. (6) is similar in form to the standard one, describing generic two-component systems. Equation $\varepsilon(k,\omega)=0$ determines the dispersion and damping of collective modes. The key quantity here is $\sigma_{s}$, which has to be found accounting for the interaction of single-particle excitations in SC layer with the Cooper pair condensate.

Let us, first, switch off the interaction between the layers and analyze the collective modes in each layer separately. Formally, the noninteracting case corresponds to putting $d\rightarrow\infty$ in Eq. (6). Then $\varepsilon(k,\omega)=0$ splits into two independent conditions,

Let us consider these cases separately.

We have already found, that the plasmon mode of SC layer lies above the boundary of the particle-hole continuum, $\omega_{n}\gg kv_{n}$, whereas the AV mode has the phase velocity $u\ll v_{s}$. Hence there exist two limiting cases, depending on the ratio between the Fermi velocities in SC, $v_{s}$, and normal 2DEG, $v_{n}$, layers (Fig. 3). Let us consider both of these scenarios independently.

Let us expose the system to an external EM field and study light-matter interaction. We will do a numerical analysis of EM power absorption $\textmd{Re}\,(j_{n}E_{0}^{*})$ [where $j_{n}$ is defined in Eq. (5)] in the case when $v_{s}>v_{n}$ and, thus $N_{n}<N_{s}$. In the opposite regime, the interaction of the modes is weak.

First, we switch off the interaction between the layers. The EM power absorption of an isolated semiconducting layer is characterized by a standard Lorenz-shaped resonance at the plasmon frequency. A more interesting situation arises in the isolated SC layer due to the presence of both the qAB and AV modes. The AV mode is quasi-neutral, and hence it interacts with the external EM field weakly, thus the EM power absorption is negligibly small. Indeed, the approximate analytical solution Eq. (12) if substituted into Eq. (9) gives $\sigma_{s}=0$ and $j_{n}=0$ reflecting the quasi-neutral nature of AV mode and nearly zero EM power absorption. The exact numerical calculation of the power absorption in the absence of interaction between SC and semiconducting layers ($d\rightarrow\infty$), presented in Fig. 6(a), supports this conclusion. At large frequencies $\omega\gg ku$, the absorption in Fig. 6(a) reaches a plateau with the value equal to the static Drude absorption of EM radiation of normal electrons in SC layer, as it is seen from Eq. (9). Indeed, at $\omega\gg ku$ the conductivity of SC layer resembles the static Drude conductivity $\sigma_{s}\approx\sigma_{s0}$. The arrow in Fig. 6(a) indicates EM power absorption at the position of AV mode which is negligibly small in comparison with the power absorption corresponding to the plateau value.

In contrast, the qAB mode strongly interacts with external EM field and demonstrates a pronounced peak of absorption with the amplitude much greater than the plateau value, as it is shown in Fig. 6(b).

Furthermore, let us switch the interlayer interaction on and consider the EM power absorption by the hybrid system when the collective modes of SC are hybridised with the plasmon mode in the semiconducting layer (Fig. 7). Figure 7(a) shows the EM power absorption by the hybrid AV-plasmon modes. Weak hybridization of the modes results in two new eigen modes. One of them almost resembles the original plasmon mode, and the other mostly possesses the properties of bare AV mode. From our previous discussion of the light absorption by the bare modes, we expect that the quasi-plasmon mode (the hybrid mode which mostly inherited the properties of the plasmon mode) will give a pronounced resonance in EM absorption spectrum, whereas the other, quasi-AV mode (the hybrid mode which mostly inherited the properties of the AV mode) will not be seen. This reasoning is supported by the exact numerical calculations, shown in Fig. 7(a). We see a single resonance at the frequency of the quasi-plasmon mode with a shifted position with respect to bare plasmon dispersion and a pronounced Lorenz shape. The quasi-AV mode is not seen in the figure.

At large $\Omega$, the EM power absorption spectrum again reaches the plateau with the value equal to the Drude conductivity of the SC layer. The amplitude of the Lorenz-shaped resonance is smaller than the value at the plateau since the total density of electrons in the semiconducting layer is smaller than the density of electrons in SC layer, $N_{n}<N_{s}$.

Finally, EM power absorption by the hybrid qAB-plasmon mode demonstrates a double-resonance structure, indicating that both the new modes formed from the original qAB and plasmon modes (before the coupling) are extremely sensitive to external EM radiation. We want to point out, that such shape of the spectrum can be attributed to the Fano resonance, which is expected in hybrid Bose-Fermi systems BKS . And again, at large frequencies, the energy absorption reaches the plateau with the value equal to the Drude conductivity absorption of SC layer.

Let us discuss the actual parameters of the sample to be used in the experiments. We will consider qAB mode first. The Drude model is applicable if $k_{F}l\gg 1$, where $l$ is the mean free path of normal electrons. We choose $N_{s}=10^{14}$ cm^-2 and $m_{s}=0.5$ $m_{0}$, which are typical for superconductors based on transition-metal dichalcogenides japans . Then $k_{F}l\approx 1.38\times 10^{15}\tau_{s}$, and thus we can take $\tau_{s}=10^{-14}$ s. Note, with these parameters the resistivity amounts to kOhm (which corresponds to experimentally measured values), while the Bohr radius becomes $a_{B}\approx\epsilon\cdot 10^{-10}$ m. For $\epsilon=6$, the dimensionless ratio $a_{B}/v_{F}\tau_{s}$ used in all the plots is equal to $1/10$, that corresponds to the interlayer distance $d=100v_{F}\tau_{s}=1000a_{B}=600$ nm. In the normal layer, we use typical parameters $N_{n}=10^{12}$ cm^-2 and $\tau_{n}=10^{-12}$ s. They give for the ratio of static conductivities $\sigma_{s0}/\sigma_{n}(\omega=0)=1$.

Second, let us consider AV mode. To bring the plasmon and AV mode dispersions together, we need a change of parameters. In our calculations, we make the ratio $N_{s}/N_{n}$ 50 times bigger. If we keep $N_{n}=10^{12}$ cm^-2 (the same as before), then the scattering times $\tau_{s}$ and $\tau_{n}$ should become $\sqrt{50}$ times smaller to keep the dimensionless ratios $\tau_{s}/\tau_{n}$ and $a_{B}/v_{F}\tau_{s}$ unchanged. Note, unlike qAB mode, AV mode starts feeling the presense of plasmons in the normal layer at an order of magnitude smaller interlayer distances, $d=10v_{F}\tau_{s}=100a_{B}=60$ nm.

In the calculations and plots, we used dimensionless wave vector and frequency, $\Omega=\omega\tau_{s}\sim 0.01$. Then $\omega/2\pi\sim 0.1-1$ THz, which corresponds to the wavelength of the order of $0.1-1$ mm. For $q=0.4$ and $q=0.006$ we find $2\pi/k\sim 1~{}\mu$m and $\sim 70~{}\mu$m, respectively, which corresponds to the period of typical diffraction gratings (which are required in the experiments to excite the plasmons).

Let us now discuss the limitations of our theoretical approach. When a normal 2DEG is in the vicinity of a 2D superconductor, there might arise several phenomena, one of which is the proximity effect RefHolm ; RefProx2 , which might result in an induced superconductivity in the semiconducting layer. In our calculations, we used a relatively large 2DEG-superconductor separation. Evidently, the smaller the separation, the more pronounced the coupling between the qAB, AV and plasmon modes. However, at very small separations, there will arise the proximity effect, which might quench the formation of collective modes and should be accounted for in engineering samples. It limits the separation distance to the coherence length of the particular superconductor. For instance, the coherence length amounts to several nanometers in the case of high-$T_{c}$ superconductors.

An electron in 2DEG with energy below $\Delta$ can tunnel into superconductor due to the Andreev scattering RefAndreev . However, if the distance between the layers is large and most of electrons are paired, the probability of such tunneling is small. If the superconductor is exposed to external light with the frequency above $\Delta$, the tunneling can be enhanced. Indeed, external EM fields excite electron and hole quasiparticles (which coexist with SC fluctuations in the superconductor). In this case, in addition to the Andreev scattering, there occurs common tunneling of quasiparticles into the superconductor. Usually, this effect can also be disregarded when studying collective modes. In particular, the frequency of external EM field which we consider in this article (to excite the modes) is smaller than $\Delta$.

We have studied analytically and numerically the coupling of collective modes in a hybrid two-dimensional electron gas–superconductor structure in the vicinity of the critical temperature of superconducting transition of the superconductor. The layer containing the electron gas in normal state can be represented by a semiconductor or a metallic layer, used in hybrid NS contacts RefAlt . We have shown, that the superconducting layer lodges two collective modes with drastically different physical properties, which results in their different sensitivity to external electromagnetic fields. As a result, these modes couple differently with the plasmon mode of the normal electron gas spatially separated from the superconducting layer. It manifests itself in different spectra of the electromagnetic power absorption of the hybrid system.