1 INTRODUCTION

Two-subband magnetotransport of electrons in GaAs quantum wells with modulated doping has been investigated experimentally for more than 40 years [1]. At the initial stage of these studies, it was found that the filling of the second subband of dimensional quantization in the GaAs quantum well leads not only to the appearance of the second series of Shubnikov-de Haas (SdH) oscillations, but also to a decrease in the mobility (μ0) of the quasi-two-dimensional electron gas [2]. In addition, GaAs quantum wells with two filled subbands of dimensional quantization E1 and E2, were found to exhibit classical positive magnetoresistance (PMR) due to the difference of mobilities in the subbands [3, 4]. It has also been shown that a new type of quantum resistance oscillations—magneto-intersubband (MIS) oscillations [5, 6] arises in two-subband electron systems.

The decrease of mobility μ0 in a single GaAs quantum well when the second subband is filled, as well as the occurrence of MIS oscillations, confirmed the fundamental role of intersubband scattering in two-subband classical and quantum magnetotransport [7, 8]. Studies of two-subband magnetotransport in double GaAs-quantum wells have also confirmed the necessity to take into account intersubband scattering when interpreting classical PMR in bilayer electron systems [9–11], but the question of the contribution of intersubband scattering to classical PMR in single-layer systems is still open. In the present work, magnetotransport has been investigated in a single-layer two-subband electron system realized on the basis of a selectively doped single GaAs quantum well with short-period AlAs/GaAs superlattice barriers [12–15]. It is found that in such structures, as in double GaAs-quantum wells, the contribution of intersubband electron scattering to the classical PMR cannot be neglected. The obtained experimental data are analyzed within the framework of known theoretical models of classical and quantum transport in two-subband electron systems.

2 MAGNETORESISTANCE OF TWO-DIMENSIONAL SYSTEMS WITH TWO FILLED SUBBANDS

At low temperatures, when the electron gas is degenerate and elastic scattering on impurities or other inhomogeneities is the main scattering mechanism, the classical PMR in a two-subband system is expressed by the relation [9, 11]

$$\frac{{{{\rho }_{{{\text{CPMR}}}}}}}{{{{\rho }_{0}}}} = \frac{{\omega _{c}^{2}{{\nu }_{s}}{\text{/}}{{\nu }_{0}} + \nu _{r}^{2}}}{{\omega _{c}^{2} + \nu _{r}^{2}}},$$
(1)

where ρ0 = \(m{{\nu }_{0}}\)/e2ns—resistance in the zero magnetic field, ns = n1 + n2, n1 and n2—electron densities in the subbands, ωc = eB/m—cyclotron frequency, B—external magnetic field, m—electron effective mass. The classical resistance increases with increasing B, starting at a value ρ0, and reaches saturation at ωc \( \gg \) \({{\nu }_{r}}\). The characteristic relaxation rates \({{\nu }_{s}}\), \({{\nu }_{r}}\) and \({{\nu }_{0}}\) are given by the following expressions:

$${{\nu }_{s}} = ({{n}_{1}}{\text{/}}{{n}_{s}})\nu _{{11}}^{t} + ({{n}_{2}}{\text{/}}{{n}_{s}})\nu _{{22}}^{t} + \nu _{{12}}^{t},$$
(2)
$${{\nu }_{r}} = ({{n}_{2}}{\text{/}}{{n}_{s}})\nu _{{11}}^{t} + ({{n}_{1}}{\text{/}}{{n}_{s}})\nu _{{22}}^{t} + 2\nu _{{12}}^{q} - \nu _{{12}}^{t},$$
(3)
$$\begin{gathered} {{\nu }_{0}} = (\nu _{{11}}^{t} + \nu _{{12}}^{q})(\nu _{{22}}^{t} + \nu _{{12}}^{q}){\text{/}}{{\nu }_{r}} \\ - \;{{(\nu _{{12}}^{q} + \nu _{{12}}^{t})}^{2}}n_{s}^{2}{\text{/}}4{{n}_{1}}{{n}_{2}}{{\nu }_{r}}, \\ \end{gathered} $$
(4)

where \(\nu _{{11}}^{t}\) and \(\nu _{{22}}^{t}\)—transport relaxation rates in the first and second (j = 1 and 2) subbands Ej, and \(\nu _{{12}}^{t}\) and \(\nu _{{12}}^{q}\)—transport and quantum (single-particle) relaxation rates at intersubband scattering.

It is convenient to transform the system of equations (2)–(4) to the form where in the left-hand side are the unknown quantities \(\nu _{{11}}^{t}\), \(\nu _{{22}}^{t}\), \(\nu _{{12}}^{t}\) and \(\nu _{{12}}^{q}\), and in the right-hand side—the characteristic relaxation rates \({{\nu }_{0}}\), \({{\nu }_{s}}\) and \({{\nu }_{r}}\) known from experiment:

$$\nu _{{12}}^{q} - \nu _{{12}}^{t} = 2({{n}_{1}}{{n}_{2}}{\text{/}}n_{s}^{2})({{\nu }_{r}} - {{\nu }_{s}}) \pm ({{n}_{1}}{\text{/}}{{n}_{s}} - {{n}_{2}}{\text{/}}{{n}_{s}})R,$$
(5)
$$\begin{gathered} \nu _{{11}}^{t} + \nu _{{12}}^{t} = ({{n}_{1}}{\text{/}}n_{s}^{{}})(1 + 2{{n}_{2}}{\text{/}}{{n}_{s}}){{\nu }_{s}} \\ - \;({{n}_{2}}{\text{/}}n_{s}^{2})({{n}_{1}} - {{n}_{2}}){{\nu }_{r}} \pm 2({{n}_{2}}{\text{/}}{{n}_{s}})R, \\ \end{gathered} $$
(6)
$$\begin{gathered} \nu _{{22}}^{t} + \nu _{{12}}^{t} = ({{n}_{2}}{\text{/}}n_{s}^{{}})(1 + 2{{n}_{2}}{\text{/}}{{n}_{s}}){{\nu }_{s}} \\ + \;({{n}_{1}}{\text{/}}n_{s}^{2})({{n}_{1}} - {{n}_{2}}){{\nu }_{r}} \mp 2({{n}_{1}}{\text{/}}{{n}_{s}})R, \\ \end{gathered} $$
(7)

where R ≡ (2/ns)[\({{\nu }_{r}}\)(\({{\nu }_{s}}\) – \({{\nu }_{0}}\))n1n2]0.5. However, to calculate the four quantities \(\nu _{{11}}^{t}\), \(\nu _{{22}}^{t}\), \(\nu _{{12}}^{t}\) and \(\nu _{{12}}^{q}\) from the system of three equations (5)–(7), knowledge of the parameters \({{\nu }_{0}}\), \({{\nu }_{s}}\) and \({{\nu }_{r}}\) obtained from data processing for classical PMR is not sufficient. One way to overcome this difficulty—to determine the value of \(\nu _{{22}}^{t}\) from the dependence of the amplitude of the MIS oscillations on l/B, which is given by the following expression [16–18]:

$$\begin{gathered} {{\rho }_{{{\text{MISO}}}}}{\text{/}}{{\rho }_{0}} = 2(\nu _{{12}}^{t}{\text{/}}{{\nu }_{0}})\exp [( - \pi {\text{/}}{{\omega }_{c}}) \\ \times \;(\nu _{{11}}^{q} + \nu _{{22}}^{q} + 2\nu _{{12}}^{q})]\cos (2\pi {{\Delta }_{{12}}}{\text{/}}\hbar {{\omega }_{c}}), \\ \end{gathered} $$
(8)

where \(\nu _{{11}}^{q}\) and \(\nu _{{22}}^{q}\)—single-particle relaxation rates in subbands E1 and E2, Δ12 = E2 – E1—the value of the intersubband energy splitting, \(\nu _{0}^{{}}\)—the total transport relaxation rate. The exponent in formula (8) is the product of the Dingle factors for the individual subbands. The formula (8) is obtained from the expansion in terms of small Dingle factors up to second order [11]. The normalized amplitude of MIS oscillations in a zero reverse magnetic field according to formula (8) will be written as ΔρΜISΟ/ρ0 = 2(\(\nu _{{12}}^{t}\)/\({{\nu }_{0}}\)). Determining this value from the Dingle plot for the MIS oscillations allows us to find out the value of \(\nu _{{12}}^{t}\).

3 EXPERIMENTAL RESULTS AND DISCUSSION

The initial heterostructure was a 26 nm wide GaAs quantum well with short-period AlAs/GaAs superlattice barriers [12, 13]. The charge carriers in the quantum well were provided by Si δ-doping. Single Si 5‑doped layers were located in AlAs/GaAs superlattice barriers on both sides of the GaAs quantum well at a distance of 29.4 nm from its boundaries. The heterostructure was grown by molecular-beam epitaxy on (001) GaAs substrate. The samples for magnetotransport measurements were Hall bridges of length L = 250 μm and width W = 50 μm equipped with top Ti/Au Schottky gates. The samples were fabricated using optical photolithography and liquid etching. The inset to Fig. 1 shows schematically the geometry of the sample.

Fig. 1.
figure 1

(a) experimental dependencies of ρxx/ρ0 on B: 1 – Vg = 0, 2—Vg = −2 V. The inset shows a Hall bridge: 1−6—ohmic contacts, 7—Schottky field gate. (b) Fourier spectra of the dependencies of ρxx/ρ0 on 1/B: 1—Vg = 0, 2–Vg = −2 V. The curves 2 are shifted upwards for clarity. (The colored version of the figure is available on-line.)

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Measurements were performed at temperatures T = 4.2 K in magnetic fields B < 2 T. The ρxx and ρxy resistances were measured in linear mode on an alternating electric current whose frequency was <1 kHz and whose amplitude did not exceed 1 μA. The Hall concentration of electrons nH was calculated from the value of ρxy in a magnetic field of 0.5 T. The mobility μ0 was calculated from the values of ρ0 and nH. The nH and μ0, values measured at the bridges before gate sputtering were 8.15 × 1015 m–2 and 119 m2/(B s), respectively. The concentrations of charge carriers in the subbands were calculated from the oscillation period of SdH: n1 ≈ 6.24 × 1015 m–2, n2 ≈ 1.91 × 1015 m–2. The E2 – E1 determined from the electron concentration difference (n1 – n2) was Δ12 ≈ 15.3 meV.

Figure 1a shows the dependencies of ρxx/p0 on B for two gate voltages (Vg). For Vg = 0 in the interval 0. 1 < B < 0. 5 Tl, MIS oscillations are observed, which coexist with SdH oscillations in stronger magnetic fields. For Vg = –2 V, only one series of SdH oscillations is observed. Figure 1b shows the results of the Fourier analysis of the dependencies of ρxx/ρ0 on 1/B. Three frequencies appear in the Fourier spectrum for Vg = 0. Two of them correspond to the frequencies of SdH oscillations (fSdH1 ≈ 12.63 T and fSdH2 ≈ 3.8 T), and the third—MIS oscillations (fMISO ≈ 8.83 T). The electron concentrations in the subbands (nj) calculated from the SchdH oscillation frequencies were: n1 ≈ 6.11 × 1015 m–2, n2 ≈ 1.84 × 1015 m–2. The E2 – E1 value determined from the fMISO frequency was: Δ12 ≈ 15 MeV. For Vg = –2 V, only one ShdH oscillation frequency is observed in the Fourier spectrum, which corresponds to the electron concentration of n1 ≈ nH.

Figure 2a shows the experimental dependences of nH, n1, n2 and ns on the gate voltage Vg. The dependences nH(Vg) and ns(Vg) have two characteristic intervals. In the interval Vg from 0 to –1 V, there is a weak change in nH and ns, and in the range from –1 to –2 V—a strong change. The nonlinear ns(Vg) dependence is due to “ejection” X of electrons from the AlAs layers adjacent to the top δ-doped layer [14]. When the gate voltage modulus |Vg| is increased in the range of variation Vg from 0 to –1 V, the amplitude of the MIS oscillations drops significantly, which is due to the decreased screening of the random scattering potential of the remote donor impurity X by electrons [19]. The gate voltage between –1.2 and –2 V does not change the concentration of X-electrons in AlAs/GaAs superlattice barriers. For this reason, in this range of Vg, the dependence of ns(Vg) is close to linear [14].

Fig. 2.
figure 2

(a) Experimental dependences of nH, n1, n2 and ns on Vg. (b) Experimental dependences of ρxx on Vg for different magnetic field values B.

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The dependences of ρxx(Vg), shown in Fig. 2b, show that the magnetic field significantly transforms them. In zero magnetic field, when |Vg| is increased, a smooth increase in ρxx is observed due to the decrease in ns and μ0. The dependence of ρxx(Vg) for B = 0.5 Tl differs significantly from ρxx(Vg) in zero magnetic field only in the range of Vg from –1.2 to –1.6 V. In this range, with increasing |Vg|, the resistance of ρxx first increases, reaches its maximum value at Vg = –1.4 V, and then decreases. This behavior of ρxx(Vg) was observed previously in a GaAs double quantum well and was explained by the manifestation of the classical PMR [11]. The dependence of ρxx(Vg) for B = 1 T qualitatively differs from ρxx(Vg) for B = 0.5 T only in the range of Vg from –1.6 to –2 V. In this range, oscillations of SdH are observed in the ρxx(Vg) dependence for B = 1 T due to the change of concentration n1 at a fixed magnetic field value. The significant quantitative difference of ρxx(Vg) in magnetic fields of 0.5 and 1 T at gate voltages of –0.9 and –1.4 V is due to the different PMR values and quantum oscillation amplitudes for these magnetic field values.

Figure 3 shows that the theory of classical PMR in a two-subband system describes well the experimental dependences only at the initial intervals of ρxx/ρ0 on B. This behaviour is due to the contribution to PMR of other classical and quantum mechanisms of electron scattering in two-dimensional systems [20–22]. It is evident that the deviation of the experimental dependences from the calculated ones starts in the magnetic fields in which MIS oscillations (Fig. 3a) and SdH oscillations (Fig. 3b) occur. This behavior of ρxx/ρ0 from B indicates that the additional contribution to the PMR has a quantum [21, 22] character. This assumption is consistent with the results of [11]. However, in our case, accounting for quantum PMR is not possible because it requires the values of \(\nu _{{11}}^{q}\) and \(\nu _{{22}}^{q}\). Nevertheless, as will be seen later, fitting the calculated dependencies of ρxx/ρ0 on B at the initial sites gives good quantitative agreement of the experiment with theory.

Fig. 3.
figure 3

Experimental dependences of ρxx/ρ0 on B at T = 4.2 K for two different values of Vg and the classical PMR calculated by the formula (1). (a) Vg = 0: \({{\nu }_{0}}\) = 23.12 GHz, \({{\nu }_{s}}\) = 24.45 GHz, \({{\nu }_{r}}\) = 92 GHz. (b) Vg = −1.4 V: \({{\nu }_{0}}\) = 27.9 GHz, \({{\nu }_{s}}\) = 73 GHz, \({{\nu }_{r}}\) = 670 GHz.

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Figure 4a shows the dependences of \({{\nu }_{0}}\), \({{\nu }_{s}}\) and \({{\nu }_{r}}\) on Vg. The values of \({{\nu }_{0}}\) were determined from the experimental values of ρ0, and \({{\nu }_{s}}\) and \({{\nu }_{r}}\)—from fitting the calculated and experimental dependences of ρxx/ρ0 on B. The increase in these relaxation rates with increasing |Vg| is primarily due to an increase in the transport relaxation rate in the second subband \(\nu _{{22}}^{t}\), since the magnitude of the Fermi vector in the second subband kF2 decreases most significantly with increasing |Vg| compared to the relatively weakly varying magnitude of kF1. The dependences of ρxx/ρ0 on Vg, shown in Fig. 4b, demonstrate good agreement of the experimental data with the calculated ones. The observed agreement allows us to consider the fitting of the calculated and experimental dependences ρxx/ρ0 on B at the initial parts of the curves as sufficiently correct.

Fig. 4.
figure 4

(a) Dependences of \({{\nu }_{0}}\), \({{\nu }_{s}}\) and \({{\nu }_{r}}\) on Vg. (b) Experimental and calculated dependence of ρxx/ρ0 on Vg for B = 0.5 T.

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Figure 5 shows the results of solving the system of equations (2)–(4) in the form of expressions (5)–(7). The dependences of (\(\nu _{{12}}^{q}\) – \(\nu _{{12}}^{t}\)) and (\(\nu _{{11}}^{t}\) + \(\nu _{{12}}^{t}\)) on Vg allow an unambiguous choice of physics-based solutions. As the upper subband empties, the intersubband scattering intensity decreases, which corresponds to a decrease in the relaxation rates \(\nu _{{12}}^{q}\) and \(\nu _{{12}}^{t}\). Therefore, the difference (\(\nu _{{12}}^{q}\) – \(\nu _{{22}}^{t}\)) should also decrease with increasing |Vg| and in the limit tend to zero. At the same time the sum of (\(\nu _{{11}}^{t}\) + \(\nu _{{12}}^{t}\)) must tend to \({{\nu }_{0}}\). We can formally trace such a limit transition if in (3), (4) and (6) we put \(\nu _{{12}}^{q}\) = \(\nu _{{12}}^{t}\) = 0. Then \({{\nu }_{r}}\) ≈ \(\nu _{{22}}^{t}\) and \({{\nu }_{0}}\) ≈ \(\nu _{{11}}^{t}\nu _{{22}}^{t}{\text{/}}{{\nu }_{r}}\) ≈ \(\nu _{{11}}^{t}\). The above behavior of (\(\nu _{{12}}^{q}\) – \(\nu _{{12}}^{t}\)) and (\(\nu _{{11}}^{t}\) + \(\nu _{{12}}^{t}\)) is observed for the second set of solutions, which corresponds to the sign “–” at R in expression (5). Therefore, only this set of solutions we consider as physically justified for describing the classical PMR in the investigated single-layer two-subband system.

Fig. 5.
figure 5

The dependences of solutions (5)−(7) on Vg: (a) (\(\nu _{{12}}^{q}\) – \(\nu _{{12}}^{t}\)); (b) (\(\nu _{{11}}^{t}\) – \(\nu _{{12}}^{t}\)) and (c) (\(\nu _{{22}}^{t}\) + \(\nu _{{12}}^{t}\)). Squares—the first set of solutions; circles—the second set of sol-utions.

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The experimental dependence of ρxx/ρ0 on B, shown in Fig. 6a, shows that in the studied system for Vg = –1 V, in addition to PMR, MIS oscillations are well manifested. The smoothed dependence of ρsm/ρ0 on B shows that it also exhibits magnetophononon oscillations [23, 24]. In accordance with expression (8), the dependence of the normalized MIS amplitude of ΔρMISO/ρ0 oscillations on 1/B on a semi-logarithmic scale is linear. Figure 6b shows good agreement between the experimental and calculated dependences of ΔρMISO/ρ0 on 1/B, which allows us to determine the value of \(\nu _{{12}}^{t}\). However, correct processing of the experimental dependences of ΔρMISO/ρ0 on 1/B to determine the value of \(\nu _{{12}}^{t}\) is possible only in a relatively narrow interval Vg near –1 V. In the Vg band preceding this interval, the experimental dependences are not linear enough and therefore are not described by the formula (8), and in the Vg band following this interval, the amplitude of MIS oscillations is strongly suppressed [25].

Fig. 6.
figure 6

(a) Experimental and smoothed dependence of ρxx/ρ0 on B. Calculation under formula (1): \({{\nu }_{0}}\) = 25.81 GHz, \({{\nu }_{s}}\) = 28.88 GHz, \({{\nu }_{r}}\) = 110 GHz. (b) Experimental dependence of ΔρMISO/ρ0 on 1/B. Calculation under formula ΔρMISO/ρ0 = 2(\(\nu _{{12}}^{t}{\text{/}}{{\nu }_{0}}\))exp[(−π/ωc)(\(\nu _{{11}}^{q}\) + \(\nu _{{22}}^{q}\) + \(2\nu _{{12}}^{q}\))]: 2(\(\nu _{{12}}^{t}\)/\({{\nu }_{0}}\)) = 0.36, (\(\nu _{{11}}^{q}\) + \(\nu _{{22}}^{q}\) + \(2\nu _{{12}}^{q}\)) = 686 GHz.

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The ΔρMISO/ρ0 value found for Vg = –1 V in a zero reverse magnetic field is 0.36. It follows that \(\nu _{{12}}^{t}\) ≈ \(0.18{{\nu }_{0}}\) ≈ 4.65 GHz. By substituting the known values \({{\nu }_{0}}\), \({{\nu }_{s}}\), \({{\nu }_{r}}\) and \(\nu _{{12}}^{t}\) into the expressions (5)–(7) for physically reasonable solutions, we can determine the transport relaxation rates of electrons in the subbands and the quantum relaxation rate for intersubband scattering, as well as the value of the ratio \(\nu _{{12}}^{q}{\text{/}}\nu _{{12}}^{t}\) at Vg = ‒1 V: \(\nu _{{11}}^{t}\) ≈ 8.8 GHz, \(\nu _{{22}}^{t}\) ≈ 87.2 GHz, \(\nu _{{12}}^{q}\) ≈ 21.5 GHz, \(\nu _{{12}}^{q}{\text{/}}\nu _{{12}}^{t}\) ≈ 4.6. In addition, the slope of the Dingle plot for the MIS oscillations (Fig. 6b) gives us the value of (\(\nu _{{11}}^{q}\) +\(\nu _{{22}}^{q}\) + \(2\nu _{{12}}^{q}\)) = 686 GHz, which allows us to estimate the value of the ratio

$$(\nu _{{11}}^{q} + \nu _{{22}}^{q}){\text{/}}(\nu _{{11}}^{t} + \nu _{{22}}^{t}) \approx \nu _{{22}}^{q}{\text{/}}\nu _{{22}}^{t} \approx 6.7.$$

Such a value of \(\nu _{{22}}^{q}{\text{/}}\nu _{{22}}^{t}\) indicates that \(\nu _{{22}}^{q}\) at Vg = ‒1 V is mainly determined by electron scattering on the remote impurity. Then for the quantum relaxation rate, we can write [26–28]:

$$\nu _{{jj}}^{q} \approx \nu _{{jj}}^{{qR}} = (\pi \hbar {\text{/}}2m)n_{R}^{*}{\text{/}}({{k}_{{{\text{F}}j}}}{{d}_{R}}),$$
(9)

where \(\nu _{{jj}}^{{qR}}\)—quantum relaxation rate on remote ionized donors in j-th subband, \(n_{R}^{*}\)—effective concentration of remote ionized donors, kFj = (2πnj)1/2, dR = (dS + dSQW/2), dS—spacer thickness, dSQW—single quantum well thickness. The relaxation rate \(\nu _{{22}}^{q}\) in the investigated structure at Vg = –1 V does not exceed 686 GHz. In this case, \(n_{R}^{*}\), calculated by formula (9), does not exceed 1015 m–2.

For a relatively dense (n > 1015 m–2) high-mobility 2D electron gas at low temperatures, the transport relaxation rate is determined by two main scattering mechanisms: scattering on distant charged impurities and on the background impurity. Therefore, the transport relaxation rate in the jth subband is expressed as

$$\nu _{{jj}}^{t} = \nu _{{jj}}^{{tR}} + \nu _{{jj}}^{{tB}},$$
(10)

where \(\nu _{{jj}}^{{tR}}\)—transport relaxation rate on remote ionized donors, and \(\nu _{{jj}}^{{tB}}\)—transport relaxation rate on charged background impurities. The intra-subband transport relaxation rate on remote ionized donors in the jth subband is given by the following relation [26–28]:

$$\nu _{{jj}}^{{tR}} = (\pi \hbar {\text{/}}8m)n_{R}^{*}{\text{/}}{{({{k}_{{{\text{F}}j}}}{{d}_{R}})}^{3}}.$$
(11)

For a concentration value of \(n_{R}^{*}\) = 1015 m–2, the corresponding value of \(\nu _{{22}}^{{tR}}\), calculated by formula (11), is 9.8 GHz. The large difference between the \(\nu _{{22}}^{{tR}}\) ≈ 9.8 GHz and \(\nu _{{22}}^{t}\) ≈ 87.2 GHz values indicates that \(\nu _{{22}}^{t}\), unlike \(\nu _{{22}}^{q}\), is not determined by scattering on the random potential of distant ionized donors alone. When Vg = –1 V, the value of \(\nu _{{22}}^{t}{\text{/}}\nu _{{11}}^{t}\) ≈ 9.9 is comparable to that of (n1/n2)1.5 ≈ 8.2. This fact allows us to consider that \(\nu _{{11}}^{t}\) and \(\nu _{{22}}^{t}\) are determined by electron scattering on two types of random potential—remote dopant impurity and background impurity, having the same dependence of transport relaxation rates on nj.

The intrasubband electron transport relaxation rate for scattering on charged background impurities with volume concentration nB can be expressed by the following relation [26–28]:

$$\nu _{{jj}}^{{tB}} \approx (m{\text{/}}2\pi {{\hbar }^{3}}){{({{e}^{2}}{\text{/}}2{{\varepsilon }_{0}}\varepsilon )}^{2}}{{n}_{B}}{\text{/}}k_{{{\text{F}}j}}^{3},$$
(12)

where ε0—electrical constant, ε—relative dielectric constant of the quantum well. At Vg = –1 V the value of \(\nu _{{22}}^{{tB}}\) = \(\nu _{{22}}^{t}\) – \(\nu _{{22}}^{{tR}}\) ≈ 77.4 GHz. The concentration corresponding to this value nB, calculated by formula (12), is 7 × 1020 m–3. Note that the nB value characterizes the quality of the structure and, unlike \(n_{R}^{*}\), is independent of the gate voltage.

For the investigated two-subband system at Vg = ‒1.5 V, the results of solving the system of equa-tions (2)–(4), represented by the expressions (5)–(7), allow the electron groups in the first and second subbands to be considered independent, since for this gate voltage \(\nu _{{12}}^{t}\) is significantly smaller than \(\nu _{{11}}^{t}\) and \(\nu _{{22}}^{t}\). Expressions (6) and (7) give the following values for the transport relaxation rates \(\nu _{{11}}^{t}\) and \(\nu _{{22}}^{t}\) for Vg = ‒1.5 V: \(\nu _{{11}}^{t}\) ≈ 25 GHz and \(\nu _{{22}}^{t}\) ≈ 2626 GHz. The obtained values of \(\nu _{{11}}^{t}\) and \(\nu _{{22}}^{t}\) are close to the values of \({{\nu }_{0}}\) ≈ 28.6 GHz and \({{\nu }_{r}}\) ≈ 2500 GHz, respectively, as they should be for two independent groups of charge carriers under condition n1 \( \gg \) n2. The magnitude of the \(\nu _{{22}}^{t}{\text{/}}\nu _{{11}}^{t}\) ≈ 100.5 ratio is approximately the same as the magnitude of the ratio (n1/n2)1.5 ≈ 82.8. Thus, it can be concluded that in the studied system at Vg = ‒1.5 V, as well as at Vg = –1 V, the relaxation rates \(\nu _{{11}}^{t}\) and \(\nu _{{22}}^{t}\) are determined by scattering on two types of random potential—remote alloying impurity and background impurity. The transport relaxation rate of \(\nu _{{11}}^{{tB}}\) at Vg = ‒1.5 V is 12 GHz. In this case, \(\nu _{{11}}^{{tR}}\) = \(\nu _{{11}}^{{tB}}\) ≈ 13 GHz.

The effective concentration of \(n_{R}^{*}\) at Vg = –1.5 V for \(\nu _{{11}}^{{tR}}\) ≈ 13 GHz is 9 × 1015 m–2. The resulting \(n_{R}^{*}\) at Vg = –1.5 V is about an order of magnitude larger than the \(n_{R}^{*}\) at Vg = –1 V. This means that when Vg = –1 V the value of \(n_{R}^{*}\) has not yet reached its maximum value, which is practically independent of Vg. However, the dependence of ns(Vg), which is close to linear in the range of Vg from –1.2 to –2 V, allows us to consider that in this range of gate voltages \(n_{R}^{*}\) takes its maximum value, which is 9 × 1015 m–2. The results show that in the interval Vg from –1.2 to –2 V the two-subband electron transport in the investigated structure is determined by scattering on the remote dopant impurity with the effective concentration of charged donors \(n_{R}^{*}\) ≈ 9 × 1015 m–2 and charged background impurity with the concentration nB ≈ 7 × 1020 m–3.

4 CONCLUSIONS

Based on a single GaAs quantum well with symmetric modulated superlattice doping, a high-mobility electronic system with two filled subbands E1 and E2 of dimensional quantization has been realized. The transition from two-subband to single-subband transport has been studied in a single-layer two-subband system by varying the magnitude of the negative voltage Vg on the Schottky gate to such a quantum well.

The dependence of the total electron concentration ns = n1 + n2 in a single GaAs-quantum well with AlAs/GaAs superlattice barriers on the gate voltage ns(Vg) is found to have two characteristic regions with different values of the average slope. The observed behavior is consistent with the results of [14] and is due to the fact that changing the voltage Vg in the range from 0 to –1.2 V changes not only ns in the single GaAs quantum well, but also the concentration of X‑electrons in the upper barrier to it, but in the range from ‒1.2 to –2 V changes only ns.

It is also found that when Vg is varied from 0 to ‒1.2 V, the amplitude of the MIS oscillations decreases. This behavior is explained by an increase in the effective concentration of remote ionized donors \(n_{R}^{*}\) due to a decrease in the concentration of X-electrons in the upper barrier [19] and a corresponding increase in the quantum relaxation rates in the subbands \(\nu _{{jj}}^{{qR}}\).

It is found that in the single-layer two-subband system, as well as in the bilayer [11] system, the classical PMR observed when a negative voltage is applied to the Schottky gate first rises and then falls. This behavior is explained by the depletion of the upper subband E2 and is consistent with the theory of classical PMR, which accounts for the role of intersubband scattering in two-subband magnetotransport [9, 11]. It is shown that the joint analysis of classical PMR and MMP oscillation amplitude provides an opportunity to estimate the concentration of charged background impurities nB and, accordingly, to evaluate the quality of high-mobility two-subband heterostructures.