We experimentally investigate phase shift gained by electromagnetic radiation transmitted through a two-dimensional electron system (2DES) on a dielectric substrate. We systematically examined the dependence of the phase shift on the radiation frequency and 2DES electron density for the GaAs semiconductor substrate. A theoretical approach was developed that found good agreement with experimental results. We demonstrate a practically achievable phase shift of $ 105 \xb0$. Obtained findings pave the way for the design of terahertz devices that can manipulate the radiation phase in a controlled and precise manner.

## INTRODUCTION

Plasmon and magnetoplasmon excitations in two-dimensional electron systems have been the focus of active research over the past $50$ years.^{1–5} In part, such a keen interest is due to the fact that starting from $0.3$ THz, the operation of semiconductor devices is beginning to be determined by the plasmonic response. Therefore, understanding the fundamentals of 2D plasmonics is essential for designing modern and future terahertz devices. At present, for example, we see the extensive development of terahertz plasmonic detectors,^{6–9} spectrometers,^{10,11} and generators.^{12–15}

Recently, a new type of plasma excitations has been discovered— the electromagnetic plasma waves.^{16–19} Physically, these waves originate from a hybrid motion of the 2D plasma under the action of a transverse electromagnetic wave that passes through the semiconductor substrate. It was found that as the electromagnetic wave passes through the 2DES, it gains a phase shift, which can be controlled by varying the electron concentration, $ n s$, or by applying an external magnetic field. This effect has been used to create terahertz (THz) phase shifters.^{20,21}

^{22–25}

^{,}

^{21}can be expressed as follows:

In the present paper, we report on the experimental study of the phase shift that electromagnetic radiation acquires as it passes through the 2DES located on a dielectric substrate. We investigate experimentally how the electron density in a 2DES and excitation radiation frequency relate to the magnitude of the phase shift. A physical model that describes the experimental data has been developed. Specifically, the theoretical model has provided insight into how the semiconductor substrate properties affect the phase shift. As a result, it is determined that a maximum phase shift of $ 180 \xb0$ can be achieved for the proposed phase shifter. Using an external magnetic field, we experimentally demonstrate a tunable phase shift of $ 105 \xb0$.

## EXPERIMENTAL METHOD

The experiments have been conducted on three structures with a high-quality 2DES, based on an Al $ 0.3$Ga $ 0.7$As/GaAs/Al $ 0.3$Ga $ 0.7$As quantum well. Two of them have a single $20$ nm-wide quantum well $200$ nm below the crystal surface—one with $ n s=2.8\xd7 10 11$ cm $ \u2212 2$ and the substrate thickness $d=468$ $\mu m$ (sample A) and the other with $ n s=1.3\xd7 10 12$ cm $ \u2212 2$ and $d=615$ $\mu m$ (sample B). The third heterostructure, sample C, contains five stacked quantum wells, each $20$ nm wide, with the upper quantum well $60$ nm below the crystal surface. In that case, the structure has the total electron density of $ n s=7.9\xd7 10 12$ cm $ \u2212 2$ and $d=625$ $\mu m$. The mobility of electrons in the 2DES measured by the transport method was $ \mu A=0.7\xd7 10 6 cm 2/Vs$, $ \mu B=120\xd7 10 3 cm 2/Vs$, and $ \mu C=60\xd7 10 3 cm 2/Vs$, respectively. The corresponding scattering times of electrons are $ \tau A\u224830$ ps, $ \tau B\u22485$ ps, and $ \tau C\u22482.6$ ps. The samples are shaped as square plates of $1\xd71$ cm $ 2$ in size. Each sample was mounted on a copper diaphragm $6$ mm in diameter and placed at the center of a superconducting solenoid inside a cryostat with optical windows. The solenoid provides a magnetic field of up to $7$ T. All measurements are taken at a fixed sample temperature of $5$ K. The quasi-optical measurement scheme used in the study is based on a Mach–Zehnder interferometer setup (see the supplementary material for details). As an excitation source, we employ a set of backward-wave oscillators covering the frequency range from $50$ to $500$ GHz. As a detector, we used a bolometer cooled down to $4.2$ K. The cryostat with the sample, oriented in the Faraday geometry, is placed in the probing arm of the interferometer. The reference arm includes a movable mirror, whose displacement is used to calculate the phase shift given the excitation wavelength. In our case, the phase shift is tuned by the application of an external magnetic field perpendicular to the sample surface.

## RESULT AND DISCUSSION

In Fig. 1(b), $\Delta \varphi $ introduced by the 2DES is displayed as a function of the electromagnetic radiation frequency. The measurements were taken for sample B. Two methods were used to acquire the data: the frequency sweep and the magnetic-field sweep. The corresponding data in the figure are plotted in black dots and red triangles accordingly. In contrast to Eq. (2), we observe that substrate strongly influences the phase shift. We also note that the phase shift reaches its maximum value in the vicinity of the Fabry–Pérot resonances of the substrate [marked by the dashed vertical lines in Fig. 1(b)]. It is worth noting that $\Delta \varphi $ at these Fabry–Pérot resonance frequencies is well described by Eq. (2), as demonstrated in Fig. 1(c).

^{27,28}Let us consider an electromagnetic wave incident perpendicular to the two-dimensional system located on a semiconductor substrate. Considering an incident electromagnetic wave with the electric field $ E in$ and magnetic induction $ H in$ approaching on a dielectric substrate of thickness $d$ and dielectric permittivity $\epsilon $, the fields of the transmitted wave can be found from the transmission matrix as follows:

Figure 2 illustrates the dependence of the maximum values of phase shift $\Delta \varphi $ on the frequency of the incident terahertz radiation. The data refer to samples A (black squares), B (blue pentagons), and C (red circles). It should be noted that the phase-shift maxima positions do not necessarily coincide with the Fabry-Pérot resonances. The solid-line curves in the figure show the behavior of $\Delta \varphi $ predicted by Eq. (2) derived for the 2DES in vacuum. According to data in Fig. 1, for samples A and B, the phase shift reaches its maximum close to the Fabry–Pérot resonances marked with dashed lines in the figure. At these resonance frequencies, where $ \omega N=N \omega d=Nc\pi / \epsilon d$ ( $N=1,2,\u2026$), the transmission matrix from Eq. (5) becomes an identity matrix. It means that at such frequencies, the substrate has no effect on the electrodynamics of the system. Thus, in the vicinity of the Fabry–Pérot resonances, the phase shift can be well described by Eq. (2), which assumes the complete absence of the substrate. Indeed, the experimental data for $\Delta \varphi $ obtained from samples A and B are consistent with the prediction of Eq. (2).

Given real experimental conditions, sweeping the parameter $ L K= m \u2217/ n s e 2$ from zero to infinity is practically impossible. In the present study, for instance, the maximum phase shift of $ 105 \xb0$ is achieved at $209$ GHz for the 2DES with $ n s=7.9\xd7 10 12$ cm $ \u2212 2$ on a GaAs substrate of $\epsilon =12.8$ and $d=625$ $\mu m$ (sample C), as marked by the blue dot in Fig. 3(c). The blue curve in the figure represents the theoretical data calculated according to Eq. (7), neglecting the 2DES dissipation in the tested structure. Thus, we see good agreement between the theory and experiment. For the practical application of our findings, we also consider the losses introduced by the 2DES structure. For example, Fig. 3(b) displays the transmission through the sample C measured at $209$ GHz. Here, the magnetic field sweep from $0$ to $6$ T results in $\Delta \varphi = 105 \xb0$ at the maximum transmission loss of $\u22128$ dB.

## CONCLUSION

In conclusion, our study employed an interferometric method to investigate the phase shift gained by the electromagnetic radiation transmitted through a two-dimensional electron system (2DES) on a dielectric substrate. Our research demonstrated that the phase shift can be actively controlled by applying an external magnetic field. We experimentally examined the dependence of the phase shift on various factors, including the electron density in the 2DES and the frequency of the radiation. Our theoretical framework, which is based on transmission matrices, effectively elucidated and accurately described the experiment. In particular, the theoretical model has provided insight into how the semiconductor substrate properties affect the phase shift. Notably, our investigations revealed that the proposed device allows for the $ 105 \xb0$ phase shift under realistic experimental conditions. The reported results hold substantial practical relevance for developing phase-shift devices operating in the terahertz frequency range.

## SUPPLEMENTARY MATERIAL

See thesupplementary material for details of the experimental setup. In particular, the scheme of the Mach–Zender interferometer is considered.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support from the Russian Science Foundation (Grant No. 19-72-30003).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**K. R. Dzhikirba:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal). **D. A. Khudaiberdiev:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal). **A. Shuvaev:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal). **A. S. Astrakhantseva:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal). **I. V. Kukushkin:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal). **V. M. Muravev:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings in this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Advances in Terahertz Source Technologies*

*Optical Waves in Crystals: Propagation and Control of Laser Radiation*