Plasma excitations in electron systems have been actively studied for more than five decades [1–4]. Doped two-dimensional (2D) systems exhibiting plasma effects in the microwave and terahertz ranges are of great fundamental interest and are used in important applications [5, 6] because of a number of unique properties of 2D plasmons distinguishing them from three-dimensional analogs: the characteristics of plasma excitations in 2D samples are determined by the electron density or a low external magnetic field B < 300 mT [7, 8]. Plasma resonances in the microwave, terahertz, and far infrared frequency ranges ω are certainly resolved only under the condition ωτ > 1, where τ is the electron relaxation time [4]. The progress of the technologies of the growth of 2D heterostructures increases the electron and hole mobilities by several orders of magnitude, which made it possible to study plasmons at lower frequencies down to the microwave range [9, 10] and is very important for both applications and fundamental studies.

The discovery of ferromagnetic (FM) p-type In1 ‒ xMnxAs and Ga1 – xMnxAs semiconductors [11] stimulated studies of the integration of magnetism in semiconductor electronics [12]. Studies of cyclotron resonance in FM Ga1 – xMnxAs samples in high magnetic fields up to 100 T showed that a very low mobility of charge carriers (μ ≤ 10 cm2/(V s)) is responsible for very broad peaks in the cyclotron resonance signal upon excitation in the infrared range [13]. Defects associated with interstitial Mn [11] in such FM semiconductors completely quench the interband luminescence, which seemingly makes it impossible to use these materials in optoelectronics. However, the authors of [12] proposed a hybrid structure: a FM layer near the quantum well (QW) with a 2D hole gas. The exchange interaction of the FM layer with holes in the QW should lead to the equilibrium spin polarization of holes (proximity effect), which can in turn affect the orientation of the magnetization M(T). The circular polarization of the optical transition in heterostructures with the QW and nearby FM Mn delta layer (δ-〈Mn〉) was detected in [14, 15], which was interpreted as the polarization of spins of holes in the QW in the exchange field of the FM layer. In this work, such FM structures with the InGaAs/GaAs QW and 〈Mn〉 δ layer separated by a narrow GaAs spacer with a thickness of dS = 3–10 nm [15–17] are studied by the ODCR method upon the absorption of infrared laser radiation. We detect a resonance magnetic field of ODCR atypically low for nonmagnetic InGaAs/GaAs heterostructures, which is attributed to the mixed magnetoplasmon cyclotron resonance of degenerate 2D holes on mesoscopic domains of the QW. The specific design of the structure including the high-density FM acceptor 〈Mn〉 δ layer, which is capable of separation and self-compensation, is responsible for the anomalously strong fluctuation Coulomb potential for charge carriers in the QW. Such separation of the QW into submicron domains with a characteristic size of ~0.2 μm is detected by low-temperature magnetic-force microscopy below the Curie temperature TC ~ 35 K of the 〈Mn〉 δ layer.

The studied structures with the InGaAs/GaAs QW and spatially nearby 〈Mn〉 δ layer were grown by the combined method of metal-organic epitaxy and laser sputtering [15, 16]. Schemes of various structures are presented in the insets of Fig. 1. Structure no. 1 (growth no. 4846) grown on the insulating i-GaAs 3°(001) substrate consisted of the 〈C〉 δ layer, the lower GaAs (10 nm) spacer, the 10-nm In0.22Ga0.78As QW, the upper GaAs spacer dS = 3.6 nm in thickness, and the GaAs cover layer ~40 nm thick. The nominal thickness of the 〈Mn〉 δ layer was QMn ~ 0.3 monolayer (ML), which corresponds to a very high surface density NMn of Mn ions (in GaAs, 1 ML = 6.3 × 1014 cm–2). Because of δ doping with 〈Mn〉 and 〈C〉 acceptor δ layers, the QW in such structures contains a 2D hole gas with a high density of ~1012 cm–2 [16, 19]. The surface density of holes in structure no. 1 at T = 5 K is pS ≈ 7 × 1011 cm–2 and their Hall mobility is μ ≈ 3000 cm2/(V s). The mobility of holes in the QW in InGaAs/GaAs structures without FM 〈Mn〉 δ layers is an order of magnitude higher [20]. Structure no. 2 (growth no. 5170) with the In0.1Ga0.9As/GaAs/δ-〈Mn〉 QW (dS = 10 nm) grown on a doped \({{n}^{ + }}\)-GaAs (001) substrate (~1017 cm–3) and a thin i-GaAs buffer layer ~3 nm in thickness (Fig. 1b) was also studied using the ODCR method. Since the substrate is doped, transport measurements in the plane and the direct determination of the density and type (n or p) of charge carriers in the QW are impossible in diode structures.

Fig. 1.
figure 1

(Color online) Photoluminescence spectra for the InGaAs/GaAs/δ-〈Mn〉 quantum well (black solid line) without and (red dashed line) with illumination by a 118.8‑μm infrared laser and (green dash-dotted line) their difference \(\Delta I(E)\) at T ≈ 2.4 K in structure (a) no. 1 at the resonance magnetic field Bmax ≈ 3.7 T and (b) no. 2 at the resonance magnetic field Bmax ≈ 1.8 T.

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In the experiment, we used the infrared line λ = 118.8 μm (ω = 1.60 × 1013 Hz) of a gas laser (EL = 10.43 meV, PL ~ 10 mW/cm2) with СО2 pump. Infrared radiation was supplied through a steel tube downward to a sample in a cryostat with liquid helium under evacuation (T ≈ 2.4 K) and was focused by a Teflon lens. A cw semiconductor laser at a wavelength of 785 nm (EL = 1.58 eV, PL ~ 10 mW/cm2) was used to excite photoluminescence. Photoluminescence was excited and collected out through two waveguides 200 μm in diameter and was detected on a monochromator with a base of 0.55 m and a liquid-nitrogen-cooled CCD camera. Special attention was paid to the coincidence of spots of optical and probe infrared radiations on the sample; to this end, a diaphragm 1 mm in diameter was placed in front of the sample. A magnetic field of B = 0–10 T was induced by a superconducting solenoid and was applied in the Faraday geometry (along the growth axis of the structures normal to the surface of the samples). A λ/4 plate and a linear polarizer in front of the sample ensured the measurement of circularly polarized (\({{\sigma }^{ + }}\) or \({{\sigma }^{ - }}\)) photoluminescence in the magnetic field of both signs.

The magnetic and magneto-optical properties of structures with the 〈Mn〉 δ layer were previously studied using the magneto-optical Kerr effect [17] and a SQUID magnetometer [18]. These studies confirmed that these structures are ferromagnetic and have a Curie temperature of TC ~ 30–35 K. Magnetic force microscopy studies of the surface and structure of the magnetic flux were performed with an AttoCube AttoDry 1000 atomic force microscope. Atomic force microscopy/magnetic force microscopy measurements were carried out in the exchange gas (helium) atmosphere at a pressure of P ~ 0.5 mbar in the temperature range from 4 to 50 K with a high accuracy of ~0.1 K. Bruker MESP silicon magnetic cantilevers coated with a CoCr layer were used as probes. The surface topography was studied in the semi-contact mode, and the structure of the magnetic flux was examined in the magnetic force microscopy mode with the switched-off feedback at a height of ~110 nm above the sample surface. The phase shift of vibrations of a magnetic cantilever was measured to obtain magnetic force microscopy images.

The spatial separation of the QW and 〈Mn〉 δ layer makes it possible not only to prevent the quenching of interband luminescence in the QW by defects associated with Mn but also to significantly increase the mobility of holes in the QW by almost three orders of magnitude because μ < 10 cm2/(V s) in Ga1 – xMnxAs solid solutions [21]. The interaction of charge carriers in the QW with the FM 〈Mn〉 δ layer is preserved, which is confirmed by the appearance of the anomalous Hall effect in these structures [16] and by the sharp magnetic field dependence of the degree of circular polarization PC(B) for the main optical transition in the QW: PC(B) increases rapidly in low fields B < 0.3 T and much more slowly in high fields (Fig. 2а). Such a fast increase in PC(B) is characteristic of the studied structures and does not occur in reference samples without the δ-〈Mn〉 layer [15, 19]. Furthermore, experiments with pulsed excitation demonstrate the dynamic appearance of the circular polarization PC(t) of the transition in the QW in structures with the 〈Mn〉 δ layer [17]: the polarization PC(t) in structure no. 1 increases almost linearly with time after the excitation of the QW by a linearly polarized laser pulse with an energy of EL = 1.503 eV below the GaAs barrier (Fig. 2b). The mechanisms of this behavior are actively discussed.

Fig. 2.
figure 2

(Color online) (а) Magnetic field dependences of the degree of circular polarization PC(B) for the main optical transition in the quantum well excited by a cw laser with EL = 1.580 eV at T ≈ 2.4 K. (b) Photoluminescence pulse of the main transition in the quantum well and its circular polarization PC(t) in structure no. 1 excited by a linear polarized laser pulse (at EL = 1.503 eV, B = 250 mT, and T ≈ 6 K).

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The detailed cyclotron resonance study of the 2D hole system in a wide range of the cyclotron frequency ω and magnetic field requires a high quality of structures and a high mobility of holes [22, 23]. Using the semiclassical estimate of the Hall mobility \(\mu \approx e\tau {\text{/}}{{m}_{h}}\) [4], one can estimate the relaxation time of the momentum of charge carriers τ in structure no. 1 under study with μ ~ 3000 cm2/(V s) (at T = 5 K) as τ ~ 0.3 ps, which is comparable with τ ~ 0.7 ps in the 10‑nm Ga0.47In0.53As/InP QW [24]. This estimate was calculated with the mass of the heavy hole in the plane of the QW mh ≈ 0.22m0 (here and below, m0 is the mass of the free electron), which was obtained in the study of electroluminescence in n-diode structures with the In0.16Ga0.84As/GaAs QW in the magnetic field [25]. To satisfy the condition ωτ > 1, which is necessary for reliable studies of ODCR in InGaAs/GaAs/δ-〈Mn〉 structures with a low mobility in the 2D hole channel, it is necessary to use frequencies ω ≥ 1013 Hz in the far infrared range. For a gas infrared laser with the λ = 118.8 μm (ω = 1.6 × 1013 Hz) line, we obtain ωτ ~ 5, which satisfies the ωτ > 1 criterion.

Figure 1 shows the photoluminescence spectra for (a) structure no. 1 grown on the \(i\)-GaAs insulating substrate and (b) structure no. 2 on the n-GaAs substrate in the resonance magnetic field for each structure (marked by arrows Fig. 3) without illumination I(E) and with infrared illumination ICR(E) and the difference spectra ΔI(E) = ICR(E) – I(E). A large spectral half-width of emission line of the QW (>10 meV) is due to a significant inhomogeneity of the composition and structure, as well as to a high concentration of charged and neutral impurities, which are characteristic of the studied structures with C and Mn δ layers and of the used metal-organic hybrid epitaxy growth technology [16]. It is seen that the absorption of infrared radiation in structure no. 1 with the 2D hole gas reduces the intensity in the entire band of the QW, similar to the p-GaAs/AlGaAs QW [9], where such a behavior was attributed to the resonance heating of holes caused by the absorption of infrared radiation. Infrared illumination of light-emitting diode structure no. 2 results in a more complex redistribution of the photoluminescence intensity: it decreases (ΔI(E) < 0) near the maximum of the emission band of the QW and increases (ΔI(E) > 0) at the violet edge (see Fig. 1b). Similar manifestations of ODCR with change in the sign of ΔI(E) were observed in undoped GaAs/AlGaAs QW [26], which were explained by the heating of photoexcited electrons with the subsequent redistribution of formed excitons between localized and itinerant states within the inhomogeneously broadened emission band of the QW.

Fig. 3.
figure 3

(Color online) Magnetic field dependences of the normalized integral intensity of the difference spectrum ODCR(B) for the quantum well (see the main text) for both structures at T ≈ 2.4 K. The arrows mark the maxima of the dependences.

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To quantitatively estimate the ODCR effect upon the absorption of infrared radiation by charge carriers in the magnetic field B, we calculated the integral of the absolute value of the difference photoluminescence spectrum normalized to the integral intensity of the QW \(\int I(E)dE\) ODCR\((B) = \int {\text{|}}\Delta I(E){\text{|}}dE{\text{/}}\int I(E)dE\) over the spectral emission range of the QW: 1.30–1.33 eV for structure no. 1 and 1.395–1.43 eV for structure no. 2 (see Fig. 1). The ODCR(B) dependences thus obtained are presented in Fig. 3. These dependences for structures nos. 1 and 2 demonstrate weakly pronounced maxima at Bmax ≈ 3.7 and 1.8 T, respectively. The cyclotron mass m* in the plane of the QW for charge carriers responsible for ODCR can be estimated from the semiclassical relation EFIR = ℏ-eBmax/m* for the cyclotron energy [4]. In particular, m* ≈ 0.042m0 in structure no. 1 and m* ≈ 0.020m0 in diode structure no. 2. In addition, it is seen in Fig. 3 that the maxima in both structures are observed against a significant, almost structureless background, which is usually attributed to the nonresonance heating of charge carriers [4].

When discussing the results, we note that the energy spectrum of holes in studied structures on the insulating substrate has a 2D character; therefore, the mass of holes is strongly anisotropic. This conclusion is confirmed by the observation of Shubnikov–de Haas oscillations in the magnetic field perpendicular to the plane of the QW and their absence in the parallel geometry [16]. It is known that the most reliable method of measuring the planar mass is the cyclotron resonance in the low-frequency microwave range [4, 9] because effects of nonparabolicity in the QW with a high density of holes make a significant contribution to the measured m* value in the far infrared range. In particular, detailed studies of ODCR in the p-type GaAs/AlGaAs QWs [22, 23] show that the passage from the microwave to infrared range with an increase in the excitation energy and, thereby, in the resonance field Bmax is accompanied by an increase in the cyclotron mass of holes m* = ℏeBmax/EFIR from m* ≈ 0.15m0 to m* ≈ 0.4m0, which is well reproduced in the calculations of Landau levels in the QW. The cyclotron mass m* increases with the field Bmax both at low densities of holes pS ≈ (0.3–0.6) × 1011 cm–2 [22] and at their high densities pS ≈ (0.7–1.6) × 1011 cm–2 [23].

It is interesting to compare the mass m* obtained in this work with the planar masses of holes m* obtained by other methods. This comparison shows that the m* values obtained in this work are much smaller than the values for structures InGaAs/GaAs reported in [27–30]. Furthermore, the value m* = 0.042m0 for structure no. 1 is noticeably smaller than the mass of the electron me ≈ 0.055m0 in the In0.22Ga0.78As triple compound in the QW layer of this structure, which was obtained from the concentration dependence \({{m}_{e}}(x)\) for InxGa1 – xAs [31], which makes the electron nature of the observed resonance doubtful. Similarly, the mass m* ≈ 0.02m0 in structure no. 2 is also significantly smaller than the mass of the electron me ≈ 0.062m0 for the In0.1Ga0.9As QW layer in this structure. For comparison, we also estimate the resonance magnetic field Bmax = m*EFIR/(ℏe) using m* values reported for similar structures and EFIR = 10.4 meV: (i) Bmax = 19.8 T for holes with m* = 0.22m0 in In0.16Ga0.84As/GaAs diodes [25], (ii) Bmax = 15.3 T for holes with m* = 0.17m0 in the In0.2Ga0.8As/GaAs QW at close pS ~ 7 × 1011 cm–2 [30], and (iii) Bmax = 4.9 T for electrons with me = 0.055m0 in the In0.22Ga0.78As/GaAs QW [31]. All three values are larger than Bmax ≈ 3.7 and 1.8 T in structures nos. 1 and 2, respectively.

To understand this difference of the experimentally determined nominal cyclotron masses m* and estimated fields Bmax from the published data for InGaAs/GaAs structures, it is necessary to consider possible factors that can decrease Bmax. In this direction, the detailed studies of dimensional magnetoplasma resonance in doped 2D structures (see [7–10] and references therein) seem important. These studies of high-quality GaAs/AlGaAs structures with a high mobility of both n- and p-type charge carriers (above 2 × 105 cm2/(V s)) show that a decrease in the sizes of the samples reduces Bmax at a fixed frequency of microwave excitation. Such a spatial restriction of the region of motion domain in the 2D structure by means of etching of circular mesas results in the mixing of their plasma and cyclotron modes and in the dependence of the frequencies of the upper (ω+) and lower (ω–) branches of the appearing magnetoplasma resonance on the diameter d of the circular mesa [8, 9]:

$${{\omega }_{ \pm }} = \pm \frac{{{{\omega }_{{{\text{CR}}}}}}}{2} + \sqrt {\omega _{{\text{P}}}^{2} + {{{\left( {\frac{{{{\omega }_{{{\text{CR}}}}}}}{2}} \right)}}^{2}}} ,$$
(1)

where ωCR = eB/m* is the cyclotron frequency in the field B and ωP is the plasma frequency of 2D charge carriers (holes in this case) with the density pS determined by the formula

$$\omega _{{\text{P}}}^{2} = \frac{{{{p}_{{\text{S}}}}{{e}^{2}}}}{{2m{\text{*}}{{\varepsilon }_{0}}\varepsilon (q)}}q.$$
(2)

Here, \(q = 2.4{\text{/}}d\) is the wave vector of the plasma excitation, \({{\varepsilon }_{0}}\) is the electric constant, and \(\varepsilon (q)\) is the effective permittivity of the medium. In the absence of screening by the gate, \(\varepsilon (q) = (\varepsilon + 1){\text{/}}2\), where ε = 12.8 is the permittivity of GaAs and 1 is the permittivity of free space. In the limit of high magnetic field B → ∞ or large mesa d → ∞, ω+ tends to the frequency ωCR = eB/m of the cyclotron resonance for the infinite QW. The second magnetoplasmon branch ω– has an edge character and, in the limit of high magnetic field B → ∞, corresponds to excitations propagating along the edge of the 2D system [8]. As the diameter of the mesa decreases, the plasma frequency ωP increases as \({{d}^{{ - 1/2}}}\), which results in the shift of the experimental maximum Bmax of the upper branch of resonance absorption ω+ toward lower magnetic fields with respect to the resonance magnetic field BCR = m*EFIR/(ℏe) for the infinite QW without mesa:

$${{B}_{{\max }}} = {{B}_{{{\text{CR}}}}} - \omega _{p}^{2}m{\text{*/}}(e\omega ) = {{B}_{{{\text{CR}}}}} - {\text{const/}}d.$$
(3)

This dependence on the diameter of the mesa d was observed in [9]. Thus, the spatial restriction leads to the mixing of the plasma and cyclotron modes and, as a result, to the dependence of the energy and magnetic field of resonance on the size of the 2D structure. According to Eq. (1), the formula m* = ℏeBmax/EFIR becomes inapplicable to calculate the cyclotron mass in such structures.

In the studied magnetic structures with the FM 〈Mn〉 δ layer, the restriction of the region of cyclotron motion of charge carriers can be due to the mesoscopic separation of the QW into small submicron domains caused by strong disorder. The experiment demonstrates not only quantum effects in these structures, which are inherent in high-quality 2D samples, but also the activation conductivity and a giant negative magnetoresistance caused by spin-dependent effects and magnetic disorder (see [16, 32] and references therein). The detailed analysis of the transport and magnetic data obtained in those works indicates the specific separation of the samples with the 〈Mn〉 δ layer in the plane: the QW is separated into domains containing the degenerate 2D hole gas. These domains of the QW with a characteristic size of ≥100 nm, which is much larger than the mean free path of holes l ~ 20 nm, are separated by narrow insulating (quasi-dielectric) layers with a characteristic thickness of ~10 nm [16]. Such a disorder in the studied structures appears because the grown 〈Mn〉 δ layer is an ultrathin (~2–3 nm) layer of the Ga1 – xMnxAs solid solution with a high Mn content of 2–6 at % caused by the strong diffusion of Mn in GaAs [16]. Manganese atoms located at Ga sites in the GaAs matrix (MnGa) serve not only as local magnetic moments but also as acceptor impurities. As a result, holes appear in the structure and their interaction with d-shell electrons of Mn is a physical reason for ferromagnetism in the GaMnAs solid solution [11]. Moreover, Mn atoms in the GaAs matrix can occupy not only acceptor sites MnGa but also defect interstitial sites MnI, where they serve as double donors, leading to the significant self-compensation of the Mn δ layer [11]. The total density of Mn ions in the δ layer NMn ≥ 6 × 1013 cm–2 at QMn ~ 0.3 ML [21] is much higher than the density of holes in the QW pS ≤ 0.4 × 1013 cm–2, which cannot completely screen in this case random fluctuations of the distribution of charged defects; as a result, the long-range fluctuation potential appears in the plane of the QW. This potential has the spatial range RC with the characteristic length of unscreened fluctuations in the range from the minimum value Rmin ~ dS to the maximum \({{R}_{{\max }}} \approx N_{{{\text{Mn}}}}^{{1/2}}{\text{/}}{{p}_{{\text{S}}}}\) [16]; fluctuations of the potential with a scale larger than Rmax are effectively screened by charge carriers (holes) in the 2D channel. The substitution of the characteristic values of NMn and pS yields Rmax ~ 100–200 nm. The amplitude of the fluctuation potential is estimated as γ ~ 10 meV [16], which exceeds the Fermi energy at typical densities of holes. Thus, the presence of the high-density close 〈Mn〉 δ layer with a strong self-compensation of donors and acceptors results in the appearance of the strong fluctuation Coulomb potential for acceptor charge carriers in the plane of the QW [16]. An important feature of the considered model is also the assuming existence of sharp spatial changes in the Mn concentration in the 〈Mn〉 δ layer and in the surrounding spacer and cover layer and the related strong inhomogeneity of their crystal and magnetic structures. According to this model, a quasi-2D magnetic phase in the form of separate submicron FM islands of the Ga1 – xMnxAs solid solution with a high Mn content (x ~ 0.05–0.1) is formed near the Mn δ layer [32]; this phase is characterized by strong fluctuations of the distribution of charged MnGa and MnI defects. In this case, holes in the QW are localized in potential wells formed by large-scale fluctuations of the Coulomb potential of these islands; i.e., the fluctuation potential in the QW correlates with the separated FM 〈Mn〉 δ layer because of the specificity of ferromagnetism in the Ga1 – xMnxAs semiconductor, where charge carriers (holes) also mediate the exchange coupling between Mn ions [11].

The mesoscopic separation of the QW into submicron domains with sizes ≥100 nm previously expected according to the analysis of magnetotransport data [16] was indeed detected in our low-temperature magnetic force microscopy study below the Curie temperature TC ~ 35 K of the 〈Mn〉 δ layer. Figure 4 presents atomic force microscopy/magnetic force microscopy data obtained at a low temperature of T = 4.7 K. Figure 4a shows the atomic force microscopy relief of the 1 × 1-μm region of structure no. 2. The height of inhomogeneities of the relief is 1–2 nm. Figures 4b and 4c demonstrate the magnetic flux distribution over the surface of the structure shown in Fig. 4a in magnetic fields B = 0 and 50 mT, respectively. It is seen in Fig. 4 that the contrast of the phase of vibrations of the magnetic cantilever has a characteristic scale of ~100–200 nm and certainly correlates with the relief. This structure of the magnetic flux is characteristic of the entire studied sample below its Curie temperature TC ~ 35 K and disappears after heating above TC. The observed sign-alternating phase contrast is attributed to the specific domain structure inherent in the study system and reflects a significant magnetic inhomogeneity of FM 〈Mn〉 δ layer.

Fig. 4.
figure 4

(Color online) (a) Relief of the 1 × 1- μm domain of structure no. 2 in the atomic force microscopy regime. (b, c) Magnetic flux structure (magnetic force microscopy regime) in the field perpendicular to the plane of the quantum well with the magnitude B = (b) 0 and (c) 50 mT at the temperature T = 4.7 K.

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In our opinion, the strong restriction of the effective area of active regions of the QW is responsible for mixed magnetoplasma resonance of degenerate 2D holes in an external magnetic field in such 100–200 nm mesoscopic regions of the QW with specific phase separation. We quantitatively estimate the energy of the mixed magnetoplasma mode ℏω+ (the sign + in Eq. (1) corresponding to the classical cyclotron resonance in the limit of high magnetic fields). Substituting the mass of holes m* = 0.22m0, which is obtained from photoluminescence data for light-emitting diodes with the In0.16Ga0.84As/GaAs QW [25], into Eq. (2) for the circular mesa with d = 100 nm in structure no. 1 with pS ≈ 7 × 1011 cm–2, we obtain the estimate ℏωP ≈ 9.2 meV; the cyclotron energy ℏωCR = ℏeB/m* is estimated as ℏωCR ≈ 2.2 meV. The estimate ℏω+ ≈ 10.3 meV obtained for the magnetoplasma mode is close to the energy of the infrared laser EFIR = 10.4 meV; this closeness is likely accidental taking into account a significant spread in the parameters, primarily the mass m*, in different structures. Nevertheless, the closeness of the estimated energies ℏω+ and EFIR can be considered as evidence of the proposed interpretation of the atypical manifestation of the cyclotron resonance in the studied FM heterostructures. The dependence of Bmax on the diameter of the mesa (Eq. (3)) also makes it possible to explain why Bmax is lower than the estimate BCR = m*EFIR/(ℏe) = 19.8 T obtained above for the resonance magnetic field of the classical cyclotron resonance of holes with m* = 0.22m0: \({{B}_{{\max }}} = {{B}_{{{\text{CR}}}}} - \omega _{{\text{P}}}^{2}m{\text{*/}}(e\omega )\) ≈ 6.1 T; although this value is still higher than the experimental value Bmax ≈ 3.7 T for structure no. 1, it is closer to the latter than BCR. In addition, the theory of the mixed magnetoplasma resonance for submicron samples should possibly be refined.

Since the cyclotron radius of charge carriers rc increases with a decrease in the magnetic field, the scale of unscreened fluctuations of the potential Rmax ~ 100–200 nm can become smaller than rc; as a result, the law of conservation of the wave vector can be violated upon the emission/absorption of photons in 2D structures [33]. For this reason, it is useful to estimate the cyclotron radius for the Landau level nearest to the Fermi level [33]:

$${{r}_{{\text{c}}}} = \sqrt {2\pi {{n}_{{2{\text{D}}}}}} \left( {\frac{{\hbar c}}{{eB}}} \right).$$
(4)

For structure no. 1 with n2D = pS ≈ 7 × 1011 cm–2, we obtain rc(Bmax) ~ 32 nm; i.e., the relation rc < Rmax is valid in a quite wide magnetic field range near Bmax in agreement with our model of the mixed magnetoplasma resonance of degenerate 2D holes in mesoscopic regions of the QW.

The doped \({{n}^{ + }}\)-GaAs substrate (n ~ 1017 cm–3) separated from the QW by the thin (a = 3 nm) i-GaAs layer (\(a \ll d\)) in diode structure no. 2, where it is impossible to determine the charge carrier density by standard methods and to estimate ωP, screens charge carriers in the QW [8]. As a result, the effective wave vector q ≈ 3.7/d changes, \(\varepsilon (q)\) is given by the expression \(\varepsilon (q) = (1 + \varepsilon {\kern 1pt} {\text{coth}}(qa)){\text{/}}2\), and the spectrum of 2D plasmons ωP(q) becomes linear [8]:

$${{\omega }_{{\text{P}}}} = \sqrt {\frac{{{{n}_{{{\text{2D}}}}}{{e}^{2}}a}}{{m{\text{*}}{{\varepsilon }_{0}}\varepsilon (q)}}} {\kern 1pt} q,$$
(5)

which is lower than the plasma frequency ωP for the unscreened case by a factor of  f ~ a/d ≪ 1. In this case, ωP ≪ ωCR and the restriction of the region of motion of charge carriers does not lead to noticeable renormalization of (increase in) the energy of the magnetoplasma excitation ℏω+ in Eq. (1), and the resonance ℏω+ ≈ ℏωCR = EFIR for typical hole masses requires higher magnetic fields Bmax > 10 T. At the same time, the peak observed at Bmax ≈ 1.8 T coincides with ODCR data for the \(1s \to 2{{p}^{ + }}\) line of the neutral D0 donor in n-doped GaAs/AlGaAs QWs for the infrared laser with λ = 118.8 μm [34]. These D0 donors are located in the doped substrate, a noticeable number of photoexcited charge carriers are apparently transferred from the substrate to the QW because of the short distance between them, and, as a result, ODCR appears in the luminescence signal of the QW.

Thus, the ODCR study of structures with the InGaAs/GaAs/δ-〈Mn〉 QW in the infrared range (λ = 118.8 μm laser line) has shown that the maximum of ODCR is significantly shifted toward low magnetic fields below the estimate for characteristic electron or hole cyclotron masses in the InGaAs/GaAs system. This manifestation of ODCR atypical of conventional (nonmagnetic) heterostructures with the InGaAs/GaAs QW can be treated as the manifestation of the mixed magnetoplasma cyclotron resonance of degenerate 2D holes in mesoscopic regions of the high-quality QW, which are due to the phase separation of the nearest acceptor 〈Mn〉 δ layer with a high concentration of Mn atoms. The special design of the structure including the high-density FM 〈Mn〉 δ layer, which is capable of separation and self-compensation, leads to the appearance of the anomalously strong fluctuation Coulomb potential for charge carriers in the plane of the QW. The separation of the QW into submicron domains with sizes of ~100–200 nm previously expected according to the analysis of magnetotransport data [16] has been detected in our low-temperature magnetic force microscopy study below the Curie temperature of the 〈Mn〉 δ layer. It has also been found that the resonance magnetic field in the structure grown on the insulating i-GaAs substrate is strongly different from that in the structure on the n‑GaAs substrate; this relation has been attributed to the screening of 2D plasmons in the latter structure. As a result, only the ODCR line corresponding to the \(1s \to 2{{p}^{ + }}\) transition on D0 donors of the doped substrate is observed in the light-emitting diode structure in the studied magnetic field range (<10 T).