Magneto-Intersubband Oscillations under the Conditions of Overlapping Landau Bands

Magneto-intersubband oscillations in a highly mobile two-subband electron system with one-dimensional periodic modulation of the potential under the conditions of overlapping Landau bands have been experimentally investigated. A significant modification of magneto-intersubband oscillations relative to the amplitude and phase is found: the amplitude decreases and the reversal of magneto-intersubband oscillations occurs in some ranges of magnetic fields. The obtained experimental data can be explained by a two-humped structure of the electron energy spectrum in Landau bands.

One of the implementations of a two-subband electron system is a quantum well (QW) with two filled quantum-confinement subbands E_j (j is the subband index), shown schematically in Fig. 1a. Two series of Landau levels (marked by 1 and 2 in Fig. 1b) arise in this quasi-two-dimensional electron system in an external perpendicular magnetic field with the magnitude B. The Landau levels successively cross the Fermi level (E_F) with an increase in B, which induces two series of Shubnikov–de Haas (SdH) oscillations. Shubnikov–de Haas oscillations are periodic relative to 1/B, and their frequencies (f_j) are determined by the electron densities in subbands (n_j): ${{f}_{j}} = h{{n}_{j}}{\text{/}}2e$. In addition to SdH oscillations, magneto-intersubband (MIS) oscillations arise in a two-subband system [1–8]. They occur at the frequency ${{f}_{{12}}} = {{f}_{1}} - {{f}_{2}}$.

Fig. 1.

(Color online) (a) Schematic of the profile of the confining potential of a quantum well with the width d_QW and two filled quantum-confinement subbands (E_i is the bottom of the ith subband, i = 1, 2, and E_F is the Fermi level). (b) Two series of Landau levels arising in the first and second subbands (ℏω_c is the energy spacing between Landau levels in each subband). (c) Schematic cross-sectional view of a one-dimensional lateral superlattice based on a GaAs quantum well with AlGaAs side barriers (a is the period of arrangement of Ti/Au strips).

Magneto-intersubband resistance oscillations are due to elastic intersubband scattering, which becomes resonant at coincidence of Landau levels of different subbands. In the two-subband system, MIS resistance oscillations are set by the relation [7]

$$\Delta {{\rho }_{{{\text{MISO}}}}}{\text{/}}{{\rho }_{0}} = {{A}_{{{\text{MISO}}}}}\lambda _{{{\text{MISO}}}}^{2}\cos (2\pi {{\Delta }_{{12}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}}),$$

(1)

where ${{\rho }_{0}} = {{\rho }_{{xx}}}(B = 0)$, ${{A}_{{{\text{MISO}}}}} = 2{{\tau }_{{{\text{tr}}}}}{\text{/}}{{\tau }_{{12}}}$, τ_tr is the transport scattering time, ${{\tau }_{{12}}}$ is the intersubband scattering time, $\lambda _{{{\text{MISO}}}}^{2} = {{\lambda }_{1}} \times {{\lambda }_{2}}$, ${{\lambda }_{j}} = \exp ( - \pi {\text{/}}{{\omega }_{{\text{c}}}}{{\tau }_{{{\text{q}}j}}})$ is the Dingle factor, τ_qj is the quantum lifetime, λ_MISO = $\exp ( - \pi {\text{/}}{{\omega }_{{\text{c}}}}\tau _{{\text{q}}}^{{{\text{MISO}}}})$, $\tau _{{\text{q}}}^{{{\text{MISO}}}} = 2{{\tau }_{{{\text{q}}1}}}{{\tau }_{{{\text{q}}2}}}{\text{/}}({{\tau }_{{{\text{q}}1}}} + {{\tau }_{{{\text{q}}2}}})$, Δ₁₂ = (E₂ – E₁), ω_c = eB/m* is the cyclotron frequency, and m* is the effective electron mass. Magneto-intersubband oscillations are not suppressed by temperature broadening of the Fermi distribution function [1]; therefore, they allow one to investigate quantum transport under the conditions where SdH oscillations do not manifest themselves [9–17].

In this paper, we report the results of studying MIS oscillations in a two-subband electron system in a one-dimensional periodic potential $V(x) = {{V}_{0}}\cos (2\pi x{\text{/}}a)$, where V₀ is the potential modulation amplitude and a is the period of the lateral potential modulation. One of the implementations of a lateral superlattice (LSL) is presented in Fig. 1c. In this case, V(x) is set by the gate voltage V_g on a series of metal strips formed on the surface of the semiconductor heterostructure. To date, electronic properties of one-dimensional LSLs based on single-subband systems have been investigated in detail [18–30]. However, many aspects of the electronic properties of LSLs based on two-subband systems have been poorly studied [31–35].

Two-subband systems with one-dimensional periodic modulation of the potential V(x) exhibit two series of commensurate oscillations of the magnetoresistance, the minima and maxima of which arise under the conditions [31, 33, 35]

$$2{{R}_{{{\text{c}}j}}}{\text{/}}a = (i - 1{\text{/}}4),$$

(2)

$$2{{R}_{{{\text{c}}j}}}{\text{/}}a = (i + 1{\text{/}}4),$$

(3)

where ${{R}_{{{\text{c}}j}}} = \hbar {{(2\pi {{n}_{j}})}^{{1/2}}}{\text{/}}eB$ (j = 1, 2) is the cyclotron radius and i is a positive integer. Within a classical model, these oscillations are caused by the commensurability between R_cj and a [21], whereas within the quantum-mechanical model, they are due to the oscillations of the Landau band widths [19, 20]. It has recently been shown that the one-dimensional periodic potential V(x) in the two-subband system induces not only commensurate oscillations but also the amplitude modulation of MIS oscillations [35]. The discovered phenomenon was explained by the role of Landau bands in two-subband quantum magnetotransport.

The one-dimensional periodic potential V(x) changes the shape of the electron energy spectrum in a two-dimensional system placed in perpendicular magnetic field B because the magnetic field lifts the degeneracy in the coordinate of the wavefunction center x₀, which leads to the formation of Landau bands [20]. The density of states under the conditions ${{V}_{0}} \ll {{E}_{{\text{F}}}} - {{E}_{j}} = {{\varepsilon }_{{{\text{F}}j}}}$ at a large number of filled Landau levels (N_j ~ ε_Fj/ℏω_c ≫ 1) can be written as follows [25]:

$$\begin{gathered} {{D}_{j}}{\text{/}}{{D}_{0}} = 1 + 2\sum\limits_{k = 0}^\infty \cos \left\{ {2\pi k\left[ {({{\varepsilon }_{{{\text{F}}j}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}}) - 0.5} \right]} \right\} \\ \times \;{{J}_{0}}(2\pi k{{V}_{{Bj}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}})\exp ( - \pi k{\text{/}}{{\tau }_{{{\text{q}}j}}}\hbar {{\omega }_{{\text{c}}}}), \\ \end{gathered} $$

(4)

$${{V}_{{Bj}}} = {{V}_{0}}{{J}_{0}}(2\pi {{R}_{{{\text{c}}j}}}{\text{/}}a),$$

(5)

where ${{D}_{0}} = m{\text{*/}}\pi {{\hbar }^{2}}$. The function ${\text{|}}{{V}_{{Bj}}}{\text{|}}$ is zero and maximal at the ${{R}_{{{\text{c}}j}}}{\text{/}}a$ values given by Eqs. (2) and (3) at $2\pi {{R}_{{{\text{c}}j}}}{\text{/}}a \geqslant 1$, respectively. Correspondingly, the Landau band width ${{\Gamma }_{{Bj}}} = 2{\text{|}}{{V}_{{Bj}}}{\text{|}}$ is zero and maximal under the conditions (2) and (3), respectively.

Figure 2a shows the dependences ${{\Gamma }_{{Bj}}} = 2{\text{|}}{{V}_{{Bj}}}{\text{|}}$ calculated by Eq. (5) for the regimes of (1) weak, (2) intermediate, and (3) strong modulation of the potential with respect to ℏω_c at B ranging from 0.1 to 0.25 T. The dependences of ${{D}_{j}}{\text{/}}{{D}_{0}}$ calculated by Eq. (4) for a fixed ε_Fj value on ε_Fj/ℏω_c are presented in Figs. 2b and 2c. We restrict the calculation to the sum of the first ten terms. The calculated dependences demonstrate the influence of V₀ and τ_qj on the behavior of ${{D}_{j}}{\text{/}}{{D}_{0}}$ in the first subband near the maximum of the function ${{\Gamma }_{{Bj}}}(B)$ with number i = 4. Figures 2b and 2c show that the spectra are two-humped and oscillating for weak (at $1{\text{/}}{{\tau }_{{{\text{q}}j}}} \ll {{\omega }_{{\text{c}}}}$) and intermediate (at $1{\text{/}}{{\tau }_{{{\text{q}}j}}} \sim {{\omega }_{{\text{c}}}}$) potential modulations, respectively. Note that maxima of ${{D}_{j}}{\text{/}}{{D}_{0}}$ oscillations under the condition $1{\text{/}}{{\tau }_{{{\text{q}}j}}} \sim {{\omega }_{{\text{c}}}}$ (Fig. 2c) occur at half-integer and integer ε_Fj/ℏω_c values for weak and intermediate potential modulation, respectively. Thus, passage from the weak potential modulation to the intermediate one is accompanied by reversal of ${{D}_{j}}{\text{/}}{{D}_{0}}$ oscillations, depending on ε_Fj/ℏω_c.

Fig. 2.

(Color online) (a) Magnetic field dependences of ${{\Gamma }_{{Bj}}} = 2{\text{|}}{{V}_{{Bj}}}{\text{|}}$ calculated from Eq. (5) for j = 1, ${{n}_{1}} = 6 \times$ 10¹⁵ m^–2, and a = 400 nm at V₀ = (1) 0.15, (2) 0.5, and (3) 1 meV. The position of the maximum for i = 4 is indicated by an arrow. The dotted line is the magnetic field dependence of $\hbar {{\omega }_{{\text{c}}}}$. (b, c) Ratio ${{D}_{j}}{\text{/}}{{D}_{0}}$ versus ε_Fj/ℏω_c for the energy subband with the index j = 1 calculated from Eq. (4) for ε_Fj = 21.13 meV: (thin lines) V₀ = 0.12 meV and τ_qj = (b) 100 and (c) 5 ps; (thick lines) V₀ = 0.54 meV and τ_qj = (b) 100 and (c) 5 ps.

Magneto-intersubband oscillations in the two-subband electron system in one-dimensional periodic potential V(x) under the conditions of N_j ~ ε_Fj/ℏω_c ≫ 1, ${{V}_{0}} \ll {{\varepsilon }_{{{\text{F}}j}}}$, and ${{\tau }_{{qj}}} \sim 1{\text{/}}{{\omega }_{{\text{c}}}}$ are determined by the relation [35]

$$\begin{gathered} \Delta {{\rho }_{{{\text{MISO}}}}}{\text{/}}{{\rho }_{0}} = {{A}_{{{\text{MISO}}}}} \times {{J}_{0}}(2\pi {{V}_{{B1}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}}) \times {{\lambda }_{1}} \\ \times \;{{J}_{0}}(2\pi {{V}_{{B2}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}}) \times {{\lambda }_{2}} \times \cos (2\pi {{\Delta }_{{12}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}}). \\ \end{gathered} $$

(6)

Here, the influence of V(x) on the amplitude and phase of MIS oscillations is taken into account by the factors ${{J}_{0}}(2\pi {{V}_{{Bj}}}{\text{/}}\hbar {{\omega }_{{\text{c}}}})$. Expression (6) predicts significant transformation of the amplitude and phase of MIS oscillations when ${{\Gamma }_{{Bj}}} \sim \hbar {{\omega }_{{\text{c}}}}$. The purpose of this study was to experimentally detect MIS oscillations under these conditions. To the best of our knowledge, MIS oscillations under the conditions of overlapping Landau bands have not been observed.

It was shown in [6, 8] that MIS oscillations with a large amplitude arise in highly mobile two-subband systems with a high electron density, based on selectively doped GaAs QWs with side superlattice AlAs/GaAs barriers, which opened wide possibilities for experimental investigation of multisubband quantum transport at large filling factors of Landau levels [9, 11, 13, 15, 16]. It was established that the application of a negative voltage on the Schottky gate in GaAs/AlAs heterostructures with modulated superlattice doping reduces not only the density and mobility of the two-dimensional electron gas [36] but also the τ_q value [37]. In addition, it was shown that an increase in V₀ in single-subband LSLs based on GaAs/AlAs heterostructures destroys electronic states with zero resistance, induced by microwave radiation in two-dimensional systems with one-dimensional periodic modulation [26].

In the context of these experimental facts, the main purpose of work [35] was to study quantum magnetotransport in two-subband LSLs with the minimum possible (for detecting commensurate oscillations) V₀  value and, accordingly, the maximum possible τ_qj  value. It was shown for the first time that commensurate oscillations coexist with MIS oscillations in two-subband LSLs with the period of one-dimensional potential modulation a = 300 nm and the amplitude V₀ = 0.2 meV. Only the mode of amplitude modulation of MIS oscillations could be investigated in this LSL. In this mode, one-dimensional periodic potential V(x) leads to an additional broadening of Landau levels and only slightly affects the spectrum of electronic states. The novelty of this study is that a radically different situation is considered in it for the first time: when the periodic potential V(x) leads to a two-humped structure of the energy spectrum, which fundamentally changes the two-subband quantum magnetotransport in the LSL.

In this study, we analyzed the behavior of MIS oscillations in a highly mobile two-subband electron system on the basis of a selectively doped GaAs/AlAs heterostructure. The initial heterostructure was a single 26-nm-wide GaAs QW with side superlattice AlAs/GaAs barriers [36–38]. Charge carriers in the QW were provided by δ doping with Si. Single Si δ‑doped layers were located on both sides of the GaAs QW at a distance of 29.4 nm from its edges. The distance from the QW center to the planar surface of the structure was 117.7 nm. The heterostructure was grown by molecular-beam epitaxy on a GaAs(100) substrate.

Investigations were performed on bridges with the width W = 50 µm and the length L = 100 µm prepared using optical photolithography and liquid etching. The schematic of the sample is shown in the inset of Fig. 3. The sample consists of two bridges with a one-dimensional lateral superlattice formed on one of them. The LSL with a period of a = 400 nm was a set of metal strips 60 µm long and 200 nm wide and was prepared by electron-beam lithography and the method of explosion of a Ti/Au two-layer metal film. The Au and Ti layer thicknesses were 40 and 5 nm, respectively.

Fig. 3.

(Color online) Magnetic field dependences of ${{\rho }_{{xx}}}{\text{/}}{{\rho }_{0}}$, measured at T = 4.2 K on the (1) reference bridge and (2) bridge with one-dimensional lateral superlattice. The positions of the maxima of commensurate oscillations for i = 1 in the first (j = 1) and second (j = 2) subbands are indicated by arrows. The schematic of the sample is shown in the inset.

Experiments were carried out at a temperature of T = 4.2 K in magnetic fields B < 2 T. The sample resistance was measured at ac current with a frequency of 733 Hz and an amplitude of no higher than 10^–6 A. In the initial heterostructure, the Hall concentration and electron mobility were ${{n}_{{\text{H}}}} \approx 8.2 \times {{10}^{{15}}}$ m^–2 and $\mu \approx 115$ m²/(V s). The lattice formation did not change n_H but only slightly reduced the mobility to $\mu \approx 104$ m²/(V s). In the lateral lattices under study, potential modulation occurred without application of voltage V_g to metal strips. A possible reason for this modulation is elastic stress arising between metal strips and the heterostructure [23].

Figure 3 shows the magnetic field dependences of ${{\rho }_{{xx}}}{\text{/}}{{\rho }_{0}}$ measured at T = 4.2 K on the (1) reference bridge and (2) one-dimensional LSL. On the reference bridge, only MIS oscillations are observed in the range 0.1 T < B < 0.6 T, whereas in magnetic fields B > 0.6 T, they coexist with SdH oscillations. Three frequencies manifest themselves in the Fourier spectrum (not shown) of the dependence of ${{\rho }_{{xx}}}{\text{/}}{{\rho }_{0}}$ on 1/B for the reference bridge. Two of them correspond to SdH oscillation frequencies (${{f}_{1}} \approx 12.8$ T and ${{f}_{2}} \approx$ 4.0 T), while the third one corresponds to MIS oscillations (${{f}_{{12}}} \approx 8.8$ T). The electron densities in the subbands calculated from the SdH oscillation frequencies were ${{n}_{1}} \approx 6.2 \times {{10}^{{15}}}$ m^–2 and ${{n}_{2}} \approx 1.9 \times {{10}^{{15}}}$ m^–2. The intersubband energy was determined from the frequency f₁₂ to be ${{\Delta }_{{12}}} \approx 15$ meV.

Since the maximum possible amplitude V₀ in the used LSL increases with the period a [26, 28], it is desirable to have the maximum possible a value to achieve the goal of this study. Magneto-intersubband oscillations in the initial heterostructure at T = 4.2 K manifest themselves only in fields B > 0.1 T, which limits the a value from above. In magnetic fields B > 0.6 T, MIS oscillations coexist with SdH oscillations and the ${{V}_{0}}{\text{/}}\hbar {{\omega }_{{\text{c}}}}$ and ε_Fj/ℏω_c values decrease, which complicates the investigation of the role of Landau bands in resonant intersubband scattering. In view of the aforesaid, for the experimental study of MIS oscillations in conditions of overlapping Landau bands, it is desirable to have $2{{R}_{{{\text{c}}j}}}{\text{/}}a \sim 1$ in a magnetic field $B \sim 0.6$ T.

Resistance oscillations in a one-dimensional LSL are most pronounced in magnetic fields 0.1 T < B < 0.6 T; the positions of their maxima $B_{{ij}}^{{\max }}$ are determined from Eq. (3), which allows one to consider them as commensurate. Magneto-intersubband oscillations are also observed in this magnetic-field range, but their amplitude is much smaller than that of MIS oscillations on the reference bridge. In the Fourier spectrum (not shown in Fig. 3) of the dependence of ${{\rho }_{{xx}}}{\text{/}}{{\rho }_{0}}$ on 1/B for the one-dimensional LSL in the range of inverse magnetic fields under study, in addition to the frequencies f₁, f₂, and f₁₂, the frequencies ${{f}_{{{\text{CO}}1}}} = 0.64$ T and ${{f}_{{{\text{CO}}2}}} = 0.36$ T (${{f}_{{{\text{CO}}j}}} = 2{{R}_{{{\text{c}}j}}}B{\text{/}}a$) corresponding to commensurate oscillations in the first and second subbands, respectively, are seen. Magnetic-field magnitudes $B_{{ij}}^{{\max }} = {{f}_{{{\text{CO}}j}}}{\text{/}}(i + 1{\text{/}}4)$, at which commensurate oscillations have maxima for i = 1, indicate that the value of a = 400 nm is optimal to solve the stated problem. For the LSL with this period, the parameter ${{R}_{{{\text{c}}j}}}{\text{/}}a = {{f}_{{{\text{CO}}j}}}{\text{/}}2B$ for B = 0.5 T is 0.64 and 0.36 in the first and second subbands, respectively.

Periodic components of the dependences of ${{\rho }_{{xx}}}{\text{/}}{{\rho }_{0}}$ on 1/B in the range of 2 T^–1 < 1/B < 8 T^–1 for the reference bridge and one-dimensional LSL obtained after subtracting the smoothed components are given in Fig. 4a. The behavior of MIS oscillations in the reference bridge is in complete agreement with Eq. (1). In this case, the dependence of the amplitude of MIS oscillations on 1/B is described by two fitting parameters ${{A}_{{{\text{MISO}}}}} = 0.4$ and $\tau _{{\text{q}}}^{{{\text{MISO}}}} = 8$ ps. In comparison with the reference bridge, the amplitude of MIS oscillations in the one-dimensional LSL is much smaller. In addition, the reversal of MIS oscillations is observed in some ranges of inverse magnetic fields. Figure 4b shows that MIS oscillations at the LSL and reference bridge are in phase in the range of 2.2 T^‒1 < 1/B < 2.6 T^–1 and in antiphase in the range of 5.6 T^–1 < 1/B < 6.2 T^–1.

Fig. 4.

(Color online) (a) Ratio $\Delta {{\rho }_{{{\text{MISO}}}}}{\text{/}}{{\rho }_{0}}$ versus 1/B for the (thin line) reference bridge and (thick line) one-dimensional lateral superlattice in a wide range of inverse magnetic fields. The dashed line is the dependence of ${{A}_{{{\text{MISO}}}}} \times \lambda _{{{\text{MISO}}}}^{2}$ on 1/B (${{A}_{{{\text{MISO}}}}} = 0.4$ and $\tau _{{\text{q}}}^{{{\text{MISO}}}} = 8$ ps). (b) Ratio $\Delta {{\rho }_{{{\text{MISO}}}}}/{{\rho }_{0}}$ versus 1/B for the (thin line) reference bridge and (thick line) one-dimensional lateral superlattice in two narrow ranges of inverse magnetic fields.

The dependences of $\Delta {{\rho }_{{{\text{MISO}}}}}{\text{/}}{{\rho }_{0}}$ on 1/B for the one-dimensional LSL calculated from Eq. (6) are shown in Fig. 5a. The periodic potential with the amplitude V₀ = 0.25 meV leads only to the amplitude modulation of MIS oscillations. In this case, the two-humped structure of the energy spectrum of Landau bands does not manifest itself, because ${{\Gamma }_{{Bj}}} \sim \hbar {\text{/}}\tau _{q}^{{{\text{MISO}}}}$. Good quantitative agreement between the calculated and experimental dependences is observed for ${{V}_{0}} =$ 0.65 meV. Figure 5b presents the dependences of ${{\Gamma }_{{Bj}}}$ on 1/B, calculated for ${{V}_{0}} = 0.65$ meV. In gray regions 1 (where ${{\Gamma }_{{Bj}}} < \hbar {{\omega }_{{\text{c}}}}{\text{/}}2$, j = 1, 2) and 2 (where ${{\Gamma }_{{B1}}} < \hbar {{\omega }_{{\text{c}}}}{\text{/}}2$ and ${{\Gamma }_{{B2}}} > \hbar {{\omega }_{{\text{c}}}}$), MIS oscillations for the reference bridge and LSL are in phase and antiphase, respectively.

Fig. 5.

(Color online) (a) Ratio $\Delta {{\rho }_{{{\text{MISO}}}}}{\text{/}}{{\rho }_{0}}$ versus 1/B for the one-dimensional lateral superlattice calculated by Eq. (6) at ${{A}_{{{\text{MISO}}}}} = 0.4$, $\tau _{q}^{{{\text{MISO}}}} = 6$ ps, ${{n}_{1}} = 6 \times {{10}^{{15}}}$ m^–2, ${{n}_{2}} = 1.9 \times {{10}^{{15}}}$ m^–2, $a = 400$ nm, and V₀ = (thin line) 0.25 and (thick line) 0.65 meV. (b) Width ${{\Gamma }_{{Bj}}}(1{\text{/}}B)$ calculated for energy subbands with indices (thick line) j = 1 and (thin line) j = 2 from Eq. (5) at ${{n}_{1}} = 6 \times {{10}^{{15}}}$ m^–2, ${{n}_{2}} = 1.9 \times {{10}^{{15}}}$ m^–2, $a = 400$ nm, and ${{V}_{0}} = 0.65$ meV. The dashed line is the dependence $\hbar {{\omega }_{{\text{c}}}}(1{\text{/}}B)$. Numbers 1 and 2 indicate the regions in which magneto-intersubband oscillations for the lateral superlattice and the reference sample are in phase and antiphase, respectively.

In region 1, maxima of ${{D}_{j}}{\text{/}}{{D}_{0}}$ oscillations at $1{\text{/}}{{\tau }_{{qj}}} \sim {{\omega }_{{\text{c}}}}$ arise at half-integer ε_Fj/ℏω_c values (as in the reference bridge). In this situation, the one-dimensional periodic potential V(x), with allowance for ele-ctron scattering from the random potential of impurities and defects, only reduces the amplitude of MIS oscillations but does not change their phase. In region 2, ${{D}_{1}}{\text{/}}{{D}_{0}}$ and ${{D}_{2}}{\text{/}}{{D}_{0}}$ oscillations have maxima at half-integer and integer ε_Fj/ℏω_c values, respectively (as in region 1). Here, V(x) not only reduces the amplitude of MIS oscillations but also changes their phase. In this case, the reversal of MIS oscillations occurs because ${{\Gamma }_{{B1}}} < \hbar {{\omega }_{{\text{c}}}}{\text{/}}2$ and ${{\Gamma }_{{B2}}} > \hbar {{\omega }_{{\text{c}}}}$ in region 2.

Note that the found reversal of MIS oscillations substantially differs from that of SdH oscillations in the one-dimensional LSL [24]. Shubnikov–de Haas oscillations in the two-dimensional electron gas are due to a high density of states near Landau levels. With an increase in ℏω_c, regions with the high density of states periodically cross E_F, which induces SdH oscillations. The one-dimensional periodic potential lifts the degeneracy of Landau levels, which leads to van Hove singularities in the density of states and formation of Landau bands. The density of states has maximum and minimum values at the edges and center of Landau bands, respectively [22]. This “splitting” of Landau levels in the one-dimensional periodic potential leads to splitting of maxima of SdH oscillations and their reversal under the conditions ${{\Gamma }_{{Bj}}} \sim \hbar {{\omega }_{{\text{c}}}}$ and $1{\text{/}}{{\tau }_{{{\text{q}}j}}} \sim {{\omega }_{{\text{c}}}}$ [24].

In the two-subband electron system, resonant constant-energy electronic transitions between Landau levels of different subbands occur periodically with a change in B, which induce MIS oscillations. The resonant character of these intersubband transitions is not related to the position of E_F, which makes a radical difference between the mechanisms of occurrence of MIS and SdH oscillations. Maxima of MIS oscillations occur when the Landau levels of different subbands coincide with each other. The one-dimensional periodic potential significantly transforms the conditions of the occurrence of resonant magneto-intersubband transitions. The reason is a change in the spectrum of energy states in subbands. In this case, the conditions for resonant magneto-intersubband transitions are satisfied only in some magnetic-field ranges, which was observed experimentally.

To summarize, magneto-intersubband oscillations under the conditions of overlapping Landau bands have been experimentally investigated in a fabricated one-dimensional lateral superlattice based on a highly mobile two-subband electron system. The reversal of magneto-intersubband oscillations in some magnetic-field ranges has been detected. It has been shown that reversal of magneto-intersubband oscillations occurs when the width of Landau bands in the first and second subbands is much smaller than the cyclotron energy and comparable with it, respectively.