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 °. Obtained findings pave the way for the design of terahertz devices that can manipulate the radiation phase in a controlled and precise manner.

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

Indeed, it has been shown that the 2DES impedance is well described by the Drude model,22–25,
Z 2 D ( ω ) = R + i ω L K , L K = m n s e 2 ,
(1)
where R = m / n s e 2 τ is the resistance per square, m is the effective electron mass, τ is the electron relaxation time, and L K is the kinetic inductance due to the collective motion of 2D electrons. Then, considering a 2DES in vacuum, the phase shift of the transmitted radiation21 can be expressed as follows:
tan Δ ϕ = ω L K R Z 0 2 R 1 + Z 0 2 R + ( ω L K R ) 2 ,
(2)
where Z 0 = μ 0 / ε 0 377 Ω is the vacuum impedance. Provided that ω L K R (which is equivalent to ω τ 1), and ω L K Z 0 R / 2 Eq. (2) can be simplified to the following form:
tan Δ ϕ = Z 0 2 ω L K .
(3)
A significant advantage of such a configuration is that the phase shift Δ ϕ can be tuned by changing the 2DES electron density, n s, or by applying an external magnetic field. However, the maximum phase shift that can be achieved in the given scheme, according to Eq. (2), is limited to 90 °. In fact, in Refs. 20 and 21, the maximum achieved phase shift was only 41 °.

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 ° can be achieved for the proposed phase shifter. Using an external magnetic field, we experimentally demonstrate a tunable phase shift of 105 °.

The experiments have been conducted on three structures with a high-quality 2DES, based on an Al 0.3Ga 0.7As/GaAs/Al 0.3Ga 0.7As 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 × 10 11 cm 2 and the substrate thickness d = 468 μ m (sample A) and the other with n s = 1.3 × 10 12 cm 2 and d = 615 μ 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 × 10 12 cm 2 and d = 625 μ m. The mobility of electrons in the 2DES measured by the transport method was μ A = 0.7 × 10 6 cm 2 / V s, μ B = 120 × 10 3 cm 2 / V s, and μ C = 60 × 10 3 cm 2 / V s, respectively. The corresponding scattering times of electrons are τ A 30 ps, τ B 5 ps, and τ C 2.6 ps. The samples are shaped as square plates of 1 × 1 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.

In order to determine the phase shift Δ ϕ introduced by a 2DES on a semiconductor substrate, it is necessary to be able to change the 2DES properties. Among the recently proposed methods, one convenient way to accomplish it is to suppress the 2DES conductivity by an external magnetic field applied perpendicular to the sample surface. Indeed, according to Ref. 26, the transmission of the right (+) and left ( ) polarized electromagnetic wave through the bare 2DES sheet, which is placed in a vacuum can be expressed as
t ± = 1 1 + Z 0 2 σ ± , σ ± = 1 R 1 1 + i ( ω ω c ) τ ,
(4)
where σ ± are the 2DES conductivity for different circular polarizations of the incident electromagnetic radiation, ω c is the cyclotron frequency. In our experiment, we place a grid polarizer at the output of the sample to filter the component of the electric field coinciding with the polarization of the incident wave. As a result, the transmission reads t = ( t + + t ) / 2. Then, at ω c τ = ( e B / m ) τ 1 and ω c ω, the total transparency is achieved, with transmission t 1. In this limit, the two-dimensional system has no effect on the magnitude and phase of the transmitted radiation. Therefore, for the structures under consideration, the phase shift caused by the 2DES presence can be determined as Δ ϕ = ϕ ( 7  T ) ϕ ( 0  T ). For example, the blue curve in Fig. 1(a) shows how the measured phase of the radiation transmitted through the sample varies depending on the applied magnetic field. In that case, the data were obtained for the 2DES with concentration n s = 1.3 × 10 12 cm 2 (sample B), at the Fabry–Pérot resonance frequency f = 78 GHz. The dashed black curve depicts the phase expected from the theoretical formulas (4). The results indicate that the phase of the transmitted radiation begins to saturate at the magnetic field of B = 1 T. Δ ϕ in the figure denotes the total phase shift caused by the presence of the 2DES.
FIG. 1.

(a) Phase measurements for the 78 GHz radiation transmitted through sample B with n s = 1.3 × 10 12 cm 2, plotted vs the applied magnetic field (blue line). The dashed black curve shows the phase expected from the theoretical formulas (4). (b) Frequency dependence of the phase shift Δ ϕ introduced by the 2DES: frequency sweep measurements (black dots) and magnetic-field sweep measurements (red triangles). Solid-line curves present results of calculations based on the transmission matrix approach [Eq. (6)]. The black curve corresponds to the dissipation-free plasma limit when Z 2 D = i ω L K, whereas the red curve takes into account the really present dissipation in the 2DES. Vertical dashed lines denote the Fabry–Pérot resonances of the substrate. (c) Phase shift Δ ϕ measured at the Fabry–Pérot resonance frequencies (red dots). The solid line shows the calculation according to (2).

FIG. 1.

(a) Phase measurements for the 78 GHz radiation transmitted through sample B with n s = 1.3 × 10 12 cm 2, plotted vs the applied magnetic field (blue line). The dashed black curve shows the phase expected from the theoretical formulas (4). (b) Frequency dependence of the phase shift Δ ϕ introduced by the 2DES: frequency sweep measurements (black dots) and magnetic-field sweep measurements (red triangles). Solid-line curves present results of calculations based on the transmission matrix approach [Eq. (6)]. The black curve corresponds to the dissipation-free plasma limit when Z 2 D = i ω L K, whereas the red curve takes into account the really present dissipation in the 2DES. Vertical dashed lines denote the Fabry–Pérot resonances of the substrate. (c) Phase shift Δ ϕ measured at the Fabry–Pérot resonance frequencies (red dots). The solid line shows the calculation according to (2).

Close modal

In Fig. 1(b), Δ ϕ 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 Δ ϕ at these Fabry–Pérot resonance frequencies is well described by Eq. (2), as demonstrated in Fig. 1(c).

To provide a complete theoretical description, which takes into account the presence of semiconductor substrate, we apply the transmission matrix method widely used in radiophysics and optics.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 ε, the fields of the transmitted wave can be found from the transmission matrix as follows:
( E in H in ) = ( cos ( q d ) i Z 0 ε sin ( q d ) i ε Z 0 sin ( q d ) cos ( q d ) ) ( E out H out ) ,
(5)
where q = ω ε / c is the wavevector of the electromagnetic wave in the substrate. If ω τ 1, a 2DES with impedance Z 2 D = i ω L K at the substrate surface leads to the matrix equation of the form
( E in H in ) = ( cos ( q d ) i Z 0 ε sin ( q d ) i ε Z 0 sin ( q d ) cos ( q d ) ) ( 1 0 1 Z 2 D 1 ) ( E out H out ) .
(6)
Finally, the phase shift of the electromagnetic radiation passed through the complex system can be determined as Δ ϕ = arg ( E out / E in ), where we need to account for the reflection from the front surface of the structure and for the fact that E out / H out = Z 0. The calculation results for Δ ϕ are shown as a function of radiation frequency by the black curve in Fig. 1(b). Furthermore, taking into consideration the dissipation, the 2DES impedance becomes Z 2 D = R + i ω L K. Using the same formalism of the transmission matrices from Eq. (6), we obtain another theoretical curve plotted by the red line in Fig. 1(b). The black and red curves are almost identical after the frequency of 100 GHz, which marks the transition of the 2DES response from dissipative to a reactive regime.

Figure 2 illustrates the dependence of the maximum values of phase shift Δ ϕ 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 Δ ϕ 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 ω N = N ω d = N c π / ε d ( N = 1 , 2 , ), 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 Δ ϕ obtained from samples A and B are consistent with the prediction of Eq. (2).

FIG. 2.

Frequency dependence of the maximum phase shift, Δ ϕ, measured for samples A, B, and C with the 2DES electron density n s = 2.8 × 10 11 cm 2 (black squares), 1.3 × 10 12 cm 2 (blue pentagons), and 7.9 × 10 12 cm 2 (red circles). The solid-line curves indicate the respective calculations based on Eq. (2). For sample C, the experiment data significantly differ from the theoretical calculation that assumes no substrate is present.

FIG. 2.

Frequency dependence of the maximum phase shift, Δ ϕ, measured for samples A, B, and C with the 2DES electron density n s = 2.8 × 10 11 cm 2 (black squares), 1.3 × 10 12 cm 2 (blue pentagons), and 7.9 × 10 12 cm 2 (red circles). The solid-line curves indicate the respective calculations based on Eq. (2). For sample C, the experiment data significantly differ from the theoretical calculation that assumes no substrate is present.

Close modal
In contrast, sample C shows a substantial deviation of measured data from the behavior predicted by Eq. (2). To understand the cause of such an unexpected result, we scrupulously examine the frequency dependence of the phase shift around the Fabry–Pérot resonance of 134 GHz, as shown in Fig. 3(a). Clearly, at the exact frequency of the Fabry–Pérot resonance, the phase shift measures 58 °, in full agreement with Eq. (2). However, the maximum of the phase shift, Δ ϕ = 105 °, appears at 150 GHz, when q d 2.25 π. Indeed, from the analysis of the transmission matrix from Eq. (6), it follows that given a purely reactive 2DES impedance, Z 2 D, that can vary from zero to infinity, the largest Δ ϕ is achieved at frequencies: ω = ( N + 1 / 4 ) ω d = ( N + 1 / 4 ) c π / ε d ( N = 1 , 2 , ). Hence, it can be expressed as
Δ ϕ = arctan ( 1 ε + ε Z 0 ω L K 2 + Z 0 ε ω L K ) .
(7)
Here, we note that as L K changes from zero to infinity, the phase-shift value goes from arctan ( ε ) to arctan ( 1 / 2 ε + ε / 2 ). Therefore, the absolute maximum of the phase shift that can be achieved with a 2DES on a substrate is written as a function of the dielectric permittivity of the substrate as follows:
Δ ϕ m = arctan ε + arctan ( 1 ε + ε 2 ) .
(8)
The calculated dependence of Δ ϕ m on ε is plotted as a red line in Fig. 3(c). The resultant data make it evident that as ε , Δ ϕ m 180 °. Structures can be stacked together to increase the phase shift even further. If the distance between the structures in the stack is a quarter of a wavelength, the phase shifts from each of the structures will be summed. By adding up the phase shifts, the technically demanded 360 degrees can be achieved.
FIG. 3.

(a) Phase shift Δ ϕ attributed to the 2DES, measured for sample C with n s = 7.9 × 10 12 cm 2 as a function of the radiation frequency (red circles). Dashed line marks the frequency ω = ( 2 + 1 / 4 ) ω d. The blue curve is the transmission of a bare substrate referenced to the dimensionless axis on the right, normalized to unity. (b) Transmission of sample C measured at 209 GHz vs the applied magnetic field. (c) Dependencies of the maximum possible phase shift (red curve) and of the maximum phase shift for the 2DES from sample C (blue curve) on the dielectric permittivity of the substrate. The blue dot indicates the maximum phase shift achieved experimentally for a GaAs substrate with ε = 12.8.

FIG. 3.

(a) Phase shift Δ ϕ attributed to the 2DES, measured for sample C with n s = 7.9 × 10 12 cm 2 as a function of the radiation frequency (red circles). Dashed line marks the frequency ω = ( 2 + 1 / 4 ) ω d. The blue curve is the transmission of a bare substrate referenced to the dimensionless axis on the right, normalized to unity. (b) Transmission of sample C measured at 209 GHz vs the applied magnetic field. (c) Dependencies of the maximum possible phase shift (red curve) and of the maximum phase shift for the 2DES from sample C (blue curve) on the dielectric permittivity of the substrate. The blue dot indicates the maximum phase shift achieved experimentally for a GaAs substrate with ε = 12.8.

Close modal

Given real experimental conditions, sweeping the parameter L K = m / n s e 2 from zero to infinity is practically impossible. In the present study, for instance, the maximum phase shift of 105 ° is achieved at 209 GHz for the 2DES with n s = 7.9 × 10 12 cm 2 on a GaAs substrate of ε = 12.8 and d = 625 μ 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 Δ ϕ = 105 ° at the maximum transmission loss of 8 dB.

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 ° phase shift under realistic experimental conditions. The reported results hold substantial practical relevance for developing phase-shift devices operating in the terahertz frequency range.

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

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

The authors have no conflicts to disclose.

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).

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

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