Consistent exploration of the frequency range of electromagnetic radiation between 0.1 and 1 THz is a key direction in modern physics, technology, and instrumentation. For example, imaging and vision systems in this frequency range will be capable of solving a whole range of applied problems in the fields of industrial nondestructive quality control [1–4], agriculture [5], and security [6–11]. Furthermore, the next-generation telecommunications networks are supposed to be built in the subterahertz frequency range [12–14]. At the same time, the development of subterahertz technologies will open the way for significant advances in fundamental science, especially in condensed matter physics [15–20], biomedicine [21–26], and astrophysics [27].

The successful development and large-scale application of subterahertz technologies call for inexpensive electromagnetic radiation detectors with a high efficiency and a sufficiently high speed. These requirements are met by detectors based on the principle of rectification of the alternating field of plasma waves excited in a two-dimensional electron channel by an incident electromagnetic wave at an artificial defect formed in the electron system [28–31]. The main parameters of these detectors, such as sensitivity and speed, are determined by the characteristics of the spectrum of plasma waves. Therefore, the study of the properties of these collective charge-density oscillations is a vital scientific problem.

Here, we investigate in detail the properties of plasma excitations in high-quality two-dimensional systems formed in SiGe/Si quantum wells. These semiconductor heterostructures have a number of unique characteristics, which, in particular, are relevant for their use in subterahertz radiation detectors. First of all, these structures are characterized by high purity and quality. For example, they feature low-temperature mobilities as high as 2 × 106 cm2/(V s), and the transport scattering times are comparable to those in GaAs heterostructures. Furthermore, SiGe/Si structures are fully compatible with the standard silicon technology, which will ensure in the future the low cost of detectors and the possibility of large-scale production. Thus, ohmic contacts can be made by ion implantation followed by activation [32]. From the fundamental point of view, the effective mass of two-dimensional electrons in such heterostructures is fairly large, so that the characteristic energy of the electron–electron interaction is dominant with respect to the kinetic energy. In this situation, electron–electron correlations significantly modify the basic properties of the electron system [33]. As a consequence, the study of the spectrum of plasma oscillations in SiGe/Si semiconductor heterostructures is a very important task.

The properties of collective plasma modes in SiGe/Si quantum wells were partially investigated in [34, 35] in experiments on doped quantum wells with typical low-temperature mobilities μ ~ 105 cm2/(V s). The key difference of our study is that plasma excitations were studied in a structure without any additional doping layer. A two-dimensional electron system (2DES) in the quantum well was formed when a voltage was applied to the top gate [36]. In such systems, the electron mobility increases by more than an order of magnitude and reaches a record-high value of 2 × 106 cm2/(V s). The spectrum of plasma excitations in such structures have not yet been investigated primarily because a sufficiently thick solid gold layer typically used as a necessary top gate to the sample completely reflects the incident radiation and, therefore, strongly complicates the excitation of plasma waves in the structures under consideration. In contrast, we use in this study special thin chromium layers whose characteristic resistance per square exceeds the impedance of free space. As a consequence, a significant part of incident radiation passes through the metallization layer. A similar approach was successfully applied, for example, in the study of electron spin resonance in GaN/AlGaN heterojunctions [37].

The studies were carried out on an undoped high-quality SiGe/Si structure grown by ultrahigh-vacuum chemical vapor deposition [36]. The quantum well with a width of w = 15 nm was formed between Si0.86Ge0.14 barrier layers. The samples were conventional Hall bars of width W = 120 μm with ohmic contacts formed by the evaporation of AuSb followed by thermal annealing. A SiO insulator layer (200 nm) was deposited on the sample surface. For the formation of the 2DES in a way that makes it possible to subsequently investigate its properties using microwave spectroscopy, a technique was developed and implemented for depositing a gate partially transparent to microwave radiation in the frequency range of 20–160 GHz by the thermal evaporation of a 7-nm chromium film on the surface of the dielectric layer. The typical resistance of the gate of that thickness was 1 kΩ, i.e., higher than the impedance of free space (377 Ω). The two-dimensional electron density can be varied in a wide range of ns = (1.6–2.5) × 1011 cm–2 by applying a voltage Vg to the gate. Typical low-temperature mobilities μ were on the order of (150–1000) × 103 cm2/(V s). The layout of the sample architecture is shown in Fig. 1d. For comparison, similar measurements were carried out on SiGe/Si structures grown by molecular beam epitaxy that included a Sb δ-doping layer; here, a quantum well of the width w = 15 nm is confined by Si0.82Ge0.18 barrier layers (Fig. 1e). Conventional photolithography was used to fabricate a Hall bar with a width W = 100 μm from this sample as well, and ohmic contacts were formed by the evaporation of AuSb followed by annealing. The density of two-dimensional electrons and their low-temperature mobility were ns = 3.3 × 1011 cm–2 and μ = 47 × 103 cm2/(V s), respectively. The measurements were carried out at temperatures T = 0.5–1.5 K in magnetic fields up to 10 T oriented perpendicular to the plane of the 2DES. For both structures, Si-containing quantum wells grown in the [001] direction, the lowest-energy valleys are the two ones having an isotropic electron effective mass m = 0.2m0 in the plane of the quantum well.

Fig. 1.
figure 1

(Color online) Magnetic field dependences of the microwave-induced change δRxx in the longitudinal resistance for several microwave frequencies for (a) the undoped gated SiGe/Si structure at ns = 1.6 × 1011 cm–2 (Vg = 0.8 V) and (b) the doped SiGe/Si structure with ns = 3.3 × 1011 cm–2. Horizontal lines show the signal level in the absence of microwave radiation. (c) Example of a fit by Eq. (1) for the undoped gated SiGe/Si structure at ns = 1.6 × 1011 cm–2 (Vg = 0.8 V) for a frequency of f = 97 GHz. (d, e) Layout of the sample architecture for (d) the undoped gated SiGe/Si structure and (e) the doped SiGe/Si structure.

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The experimental detection technique is based on the extreme sensitivity of the longitudinal magnetoresistance Rxx to the heating of the 2DES caused by the excitation of plasma waves in the system by microwave radiation [38]. Microwave radiation was supplied to the sample under study via an oversized waveguide with a 7 × 3.5-mm rectangular cross section (WR 28) and a cutoff frequency of f ≈ 15 GHz. The measurements were carried out in the microwave frequency range of f ≈ 20–160 GHz, covered by a set of generators with coupled frequency multiplication units. More details of the experimental technique can be found in our previous publications [39, 40]. In order to improve the signal-to-noise ratio, a the standard double lock-in-technique was used. Alternating current with an amplitude of 0.1–1 µA and a frequency of 2 kHz was passed through the sample and the voltage Vxx was read out from two potentiometric contacts by means of the first lock-in. The signal from its output reached the input of the second lock-in, tuned to the frequency fmod at which the amplitude of microwave radiation incident on the sample was modulated. Thus, the change δRxx in the longitudinal resistance of the sample under microwave irradiation was measured in the experiment. As far as the electron scattering mechanisms are highly sensitive to temperature, this leads to an increase in the sample resistance Rxx, and magnetoplasma excitations are manifested as peaks in δRxx observed when the magnetic field B is swept for a given microwave frequency (Figs. 1a and 1b).

Typical dependences of the change in the longitudinal resistance δRxx on the magnetic field B at microwave frequencies f = 40, 85, and 120 GHz for the undoped gated SiGe/Si structure with a two-dimensional electron density ns = 1.3 × 1011 cm–2 (Vg = 0.8 V) are shown in Fig. 1a. The peaks correspond to the excitation of plasmons in the system; horizontal lines show the signal levels in the absence of microwave radiation. Each curve features a pronounced resonance that shifts towards higher magnetic fields B with an increase in the microwave frequency f; this behavior corresponds to the excitation of the transverse magnetoplasma mode in the system. Figure 1b also shows typical dependences for the doped SiGe/Si structure without a gate (ns = 3.3 × 1011 cm–2) at f = 55, 90, and 155 GHz, which also demonstrate the dependence characteristic of cyclotron magnetoplasma excitation.

To determine the position of the resonance and its width ΔB, the experimental data were fitted by a theoretical dependence (see Fig. 1c).The shape of the resonance is related to the dissipative part of the longitudinal conductivity as δRxx ∝ Reσxx; in turn, the latter is described with a high accuracy by a Lorentzian dependence predicted by the Drude model:

$${\text{Re}}{{\sigma }_{{xx}}}(\omega ,B) = \frac{{{{\sigma }_{0}}}}{{2{{\tau }^{2}}}}\sum\limits_ \pm \frac{1}{{{{{(\omega \pm {{\omega }_{{{\text{mp}}}}}(B))}}^{2}} + 1{\text{/}}{{\tau }^{2}}}}.$$
(1)

Here, σ0 is the Drude conductivity, τ is the transport scattering time, and ωmp(B) is the frequency of the magnetoplasma mode in the 2DES; its square is given by the expression [41]

$${{\omega }_{{{\text{mp}}}}}{{(B)}^{2}} = \omega _{{\text{p}}}^{2} + \omega _{{\text{c}}}^{2}.$$
(2)

Here, ωc is the cyclotron frequency and ωp is the plasma frequency in zero magnetic field, which f-ollows the two-dimensional plasmon dispersion relation [42]

$${{\omega }_{{\text{p}}}} = 2\pi {{f}_{{\text{p}}}} = \sqrt {\frac{{{{n}_{{\text{s}}}}{{e}^{2}}}}{{2{{m}_{{\text{p}}}}{{\varepsilon }_{0}}\varepsilon {\text{*}}}}q} ,$$
(3)

where \(\varepsilon \text{*} = ({{\varepsilon }_{{{\text{Si}}}}} + 1){\text{/}}2\) is the effective permittivity. For a 2DES shaped as a stripe of the width W, the wave vector assumes the values q = πN/W (N = 1, 2, …). More accurate calculations yield ωp = 0.85(2πfp) for N = 1 [43].

Using the experimental data obtained in the frequency range of f ≈ 20–160 GHz, the position of magnetoplasma resonance in the δRxx(B) curves was plotted as a function of the microwave frequency for both the undoped gated SiGe/Si structure with ns = 1.6 × 1011 cm–2 at Vg = 0.8 V and the doped SiGe/Si structure with ns = 3.3 × 1011 cm–2 (see Figs. 2a and 2b, respectively). Black solid lines in these figures show the cyclotron resonance frequency ωc = eB/m for the mass m = 0.2m0. Green solid lines show the theoretical fits by Eq. (2). For the ungated doped SiGe/Si structure (Fig. 1b), the resonance positions follow the cyclotron resonance line ωc; deviation in the low-frequency region is due to the depolarization plasma shift. The theoretical fit by Eq. (2) yields a value of fp = ωp/2π = 41 GHz and the corresponding effective mass mp = (0.26 ± 0.01)m0. The latter value differs noticeably from the cyclotron mass, with the discrepancy exceeding the experimental error. Apparently, this effect is caused by the strong electron–electron interaction in the system. Indeed, a similar discrepancy was observed, for example, in narrow AlAs quantum wells [39, 44], where the characteristic energy of the electron–electron interaction is dominant with respect to the kinetic energy owing to a large effective mass.

Fig. 2.
figure 2

(Color online) Magnetodispersion of plasma excitations for (a) the undoped gated SiGe/Si structure at ns = 1.6 × 1011 cm–2 (Vg = 0.8 V) and (b) the doped SiGe/Si structure with ns = 3.3 × 1011 cm–2. Green lines show the theoretical fits by Eq. (2). Black straight lines show the cyclotron resonance frequency for a mass of m = 0.2m0. (c, d) Squared frequency of magnetoplasma excitations versus the squared magnetic field strength for (c) the undoped gated SiGe/Si structure at ns = 1.6 × 1011 cm–2 (Vg = 0.8 V) and (d) the doped SiGe/Si structure with ns = 3.3 × 1011 cm–2.

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In contrast, for the structure with a gate (Fig. 2a), the magnetoplasmon resonance strictly follows the cyclotron resonance line in the entire frequency range under study. For comparison, we show in the same figure the theoretical magnetodispersion curve calculated by Eq. (2) with ωp = 27 GHz, determined for the corresponding values of the density ns and stripe width W. It can be seen that the experimental data do not level off at this frequency. This fact clearly indicates that plasma oscillations are excited in a 2DES partially screened by a metal gate.

For a system with a metal gate having infinitely high conductivity, the plasma frequency in the limiting case of qd ≪ 1 is given by the expression [45]

$${{\omega }_{{{\text{sc}}}}} = \sqrt {\frac{{{{n}_{{\text{s}}}}{{e}^{2}}d}}{{{{\varepsilon }_{0}}\varepsilon {\text{*}}m{\text{*}}}}} q.$$
(4)

In the case under study, where the gate conductivity is finite, the use of this formula is not quite applicable, but it still gives a qualitative estimate for the frequency of the screened plasmon. Since the spatial inhomogeneity of the external electromagnetic field is weak, the wave vector q is related to the stripe width W = 120 μm by q ≈ π/W. Substituting the spacing between the 2DES and the metal gate d = 700 nm and the density ns = 1.6 × 1011 cm–2, we then obtain fsc = ωsc/2π = 5 GHz. This value (marked by the arrow in Fig. 2a) lies outside the investigated frequency range because we use a waveguide with a cutoff frequency of 15 GHz.

In addition, the cyclotron effective masses were determined from the slope of the straight line representing the dependence of the squared resonance magnetic field on the squared microwave frequency. The plots are shown in Figs. 2c and 2d. The values obtained for the two structures agree with a high accuracy and are equal to mc = (0.20 ± 0.01)m0.

The complete experimental dependence of the cyclotron effective mass mc of two-dimensional electrons on their density ns is shown in Fig. 3. The figure presents the results for several samples, i.e., for the undoped SiGe/Si structure with a gate at various voltages Vg and corresponding values of ns (red diamonds) and for the doped SiGe/Si structures investigated in this study (blue circle) and in [35] (green square). One can see that the cyclotron mass is almost independent of the electron density. It is noteworthy that, in earlier studies of magnetoplasma resonances in two-dimensional systems based on ZnO/MgZnO heterojunctions (where m = 0.33m0) [46], an appreciable increase in the effective mass with increasing electron density was observed, and this was attributed to the impact of band nonparabolicity under the conditions of strong Coulomb interaction. Similar behavior was observed in narrow AlAs quantum wells (m = 0.2m0) [39, 44]. Therefore, the independence of the cyclotron mass in SiGe/Si quantum wells, where the effective mass is comparable to that in ZnO/MgZnO heterojunctions and narrow AlAs quantum wells, from the two-dimensional electron density is an important experimental result.

Fig. 3.
figure 3

(Color online) Experimental cyclotron effective mass of two-dimensional electrons versus their density ns. Red rhombs and blue circle correspond to the data obtained for the undoped SiGe/Si structure under different gate voltages Vg and for the doped SiGe/Si structure, respectively. The green square shows the result obtained in [35].

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Let us now analyze the magnetoplasma resonance line width. We emphasize that this characteristic is of exceptional importance in the context of using SiGe/Si heterostructures as plasmonic detectors of subterahertz radiation. Since the lateral size of the mesa (about 100 μm) in the investigated samples is much smaller than the wavelength of electromagnetic radiation interacting with the plasmons, the radiative contribution to the linewidth is negligible [47, 48]. Then, the resonance width is determined solely by the quantum scattering time \({{\tau }_{q}} \sim \int {{W}_{{kk'}}}dk{\kern 1pt} '\) (here, \({{W}_{{kk'}}}\) is the probability of scattering of an electron near the Fermi surface) and may differ significantly from the transport scattering time \({{\tau }_{{{\text{tr}}}}} \sim \int {{W}_{{kk{\text{'}}}}}(1 - \cos \theta )dk{\kern 1pt} '\), where \(\theta \) is the scattering angle. Therefore, τq ≈ τcr, where \({{\tau }_{{{\text{cr}}}}} = 1{\text{/}}2\pi \Delta f\). As a rule, the quantum scattering time does not exceed the transport time, but, as shown in [48], τq becomes longer than τtr in certain cases (for example, under conditions of strong localization). The characteristic values of the transport scattering time determined from the longitudinal magnetoresistance in zero magnetic field, as well as the characteristic time τcr determined by the width of the resonance line, are given in Table 1. Although the values of τtr in the doped structure and the undoped structure with a gate differ by an order of magnitude, the difference in the width of the plasma lines is not so significant, which is clearly seen in Fig. 4. We also note that, for the structure with a gate, the ratio τtr/τcr is noticeably greater than 1, while the quantum and transport times in the doped structure are of the same order. This may be indicative of significantly poorer homogeneity of the electron system in the doped SiGe/Si heterostructure.

Table 1. Relaxation times in the investigated SiGe/Si structures
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Fig. 4.
figure 4

(Color online) Magnetic field dependences of the change δRxx in the longitudinal resistance at a microwave frequency of f = 100 GHz for (red line) the undoped SiGe/Si structure with ns = 2.4 × 1011 cm–2 (Vg = 1.3 V) and τtr = 50 ps and (blue line) the doped SiGe/Si structure with ns = 3.3 × 1011 cm–2 and τtr = 6 ps.

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In conclusion, a detailed study of plasma and magnetoplasma excitations in high-quality undoped two-dimensional systems based on SiGe/Si quantum wells has been carried out for the first time. The two-dimensional electron system has been formed by applying a voltage to the top gate that is partially transparent to microwave radiation in the frequency range of 20–160 GHz. For comparison, the results for SiGe/Si quantum wells with a Sb δ-doping layer have also been given. The transport and cyclotron relaxation times for both structures have been directly determined. It has been found that the electron effective mass is only very weakly dependent on the two-dimensional electron density in a wide range of values.