Giant Spatial Redistribution of Electrons in a Wide Quantum Well Induced by Quantizing Magnetic Field

In samples of field-effect transistors based on GaAs/AlGaAs heterostructures with an electron system in a single 50-nm-wide GaAs quantum well, a transition stimulated by a quantizing magnetic field has been detected from a bilayer state of the system in zero magnetic field to a single-layer state when only the lowest Landau level is filled. In contrast to the results for the 60-nm-wide quantum well obtained in [S. I. Dorozhkin, A. A. Kapustin, I. V. Fedorov, V. Umansky, and J. H. Smet, Phys. Rev. V 102, 235307 (2020)], the single-layer state is observed not only in incompressible quantum Hall effect states of the electron system at filling factors of 1 and 2, but also in compressible states between these filling factors. The spatial location of the single-layer system in the quantum well has been established; it appears to be independent of the electron distribution over the layers in a low magnetic field. A possible qualitative explanation for this observation has been proposed. The detected transition is supposedly due to the negative compressibility of two-dimensional electron systems caused by exchange-correlation contributions to the electron−electron interaction.

The redistribution of electrons between the layers of a bilayer electron system induced by a transverse magnetic field at large filling factors of Landau levels is well established. It occurs in bilayer systems in both double [1, 2] and wide [3] quantum wells and can be described in the Hartree approximation (see, e.g., [4] and references therein). This redistribution is caused by magnetic oscillations of the chemical potential in the layers, which are asynchronous in the case of different electron densities in them. Such a mechanism usually leads to a partial change in the electron density in the layers, keeping the bilayer state of the electron system. Applying an original capacitance technique, we have recently detected [5] reentrant transitions between bilayer and single-layer states of the electron system induced by the magnetic field in samples with 60-nm-wide single quantum wells. Single-layer states arose in states of the integer quantum Hall effect, which appeared when the total electron density in the electron system was equal to the density of single-electron states on one or two spin sublevels, i.e., at total filling factors ν = 1 and 2. In the existing theories of the quantum Hall effect in bilayer systems, such a situation can occur under the formation of coherent interlayer states (see review [6]) or the localization of electrons in one of the layers [7]. In both cases, the effect is due to the exchange and correlation contributions to the electron−electron interaction. The center of gravity of the many-body wavefunction is located between the layers in the former case and in one of the layers in the latter case. The effect of the transverse magnetic field on the distribution of the electron density in the direction of the growth axis of heterostructures in nominally bilayer electron systems was not studied experimentally before the publication of work [5]. We point only to the simulation of such a redistribution in [8], which was used to explain the existence of states of the quantum Hall effect at filling factors of 1 and 2 in a wide region of the experimental parameters.

In this work, the transition from the bilayer state in zero magnetic field to the single-layer one at the total filling factor ν ≤ 2, which corresponds to the concentration of all electrons in one layer at any experimental relation between the electron densities in the layers, is detected in samples with the width of 50 nm of the quantum well, smaller than the value of 60 nm in [5]. This observation demonstrates the existence of the magnetic-field-induced single-layer state not only in states of the quantum Hall effect but also in compressible states of the electron system, which has not yet been considered. The magnetic-field-induced transition to the single-layer state, as well as the observed asymmetry in the distribution of the electron density in the quantum well, is qualitatively explained.

Experiments were carried out with two samples fabricated from the same GaAs/AlGaAs heterostructure wafer grown by molecular beam epitaxy. The electron system in the studied samples was formed in the 50-nm-wide GaAs quantum well. The profile of the potential well and its filling with electrons were varied by means of two gate voltages applied between gates located on different sides of the quantum well and the doped contact to the electron system (see Fig. 1a). The bottom gate (BG) was formed in the process of growth of the heterostructure in the form of a heavily doped GaAs layer. The top thin-film Schottky gate (TG) was deposited on the surface of the heterostructure and had an area of S = 0.4 mm². It covered about 95% of the area of the electron system. The distance between the bottom (top) gate and the bottom (top) wall of the quantum well was d_bg = 1000 nm (d_tg = 120 nm). At zero gate voltages, a two-dimensional electron system (TL) was formed near the top of the quantum well because of electrons coming from the selectively doped layer (DL), which was created in the barrier AlGaAs layer behind a 91-nm-wide spacer located above the quantum well. The second two-dimensional electron layer (BL) was formed near the bottom of the quantum well under the application of a positive bottom gate voltage, as shown in Fig. 1a. The results reported in this work were obtained at zero voltage on the top gate (V_tg = 0). The samples had a lateral Hall bar geometry. The capacitances between both gates and the electron system, as well as the resistance and Hall resistance of the sample, could be measured simultaneously. The experimental method was described in detail in [5, 9], including a method for the determination of parasitic capacitances. The measurements of capacitances between different gates and the electron system make it possible to evaluate the electron distribution across the quantum well and to determine the electron density in the layers from the period of quantum magnetic oscillations of the capacitance [9]. The measurements of intersubband resistance oscillations [10, 11], which are observed in low magnetic fields and at comparatively high temperatures (4.2 K in our experiment), allowed us to determine the difference between the electron densities in two quantum-confined subbands. The experiments were carried out in the temperature range of 0.5 to 4.2 K in a ³He cryostat. The magnetic field up to 11 T was generated by a superconducting solenoid and was perpendicular to the plane of the quantum well.

Fig. 1.

(a) Sketch of the field effect transistor at V_bg > V_th with two layers of electrons (TL and BL) located near the opposite walls of the quantum well (WQW) and with the layer of positively charged donors DL. (b) Capacitances C_bg and C_tg between the electron system and the (black solid line, left scale) bottom and (gray solid line, right scale) top gates versus the bottom gate voltage V_bg measured in zero magnetic field at the top gate voltage V_tg = 0 and the temperature T = 1.5 K. (Empty circles, triangles) Capacitances C_bg determined from magnetocapacitance curves (see Fig. 3 and main text) in zero magnetic field and at the filling factor of Landau levels ν = 3/2, respectively. The filled circle is the capacitance calculated by Eq. (1) for ν = 3/2 at V_bg = 0. The arrow marks the gate voltage V_th ≈ 0.65 V corresponding to the formation of the bottom electron layer.

The experimental indications of the filling of the second electron layer in zero and low magnetic fields are presented in Figs. 1b and 2. These are primarily jumps in the capacitances between the gates and the electron system at V_bg = V_th seen in Fig. 1b. The change in the distance Δd_bg from the bottom gate to the layer serving as the second plate of the corresponding parallel plate capacitor is directly determined from the jump δC_bg under the assumption of a constant permittivity of the heterostructure: $\Delta {{d}_{{{\text{bg}}}}} = - {{d}_{{{\text{bg}}}}}\delta {{C}_{{{\text{bg}}}}}{\text{/}}{{C}_{{{\text{bg}}}}}$ ≈ 25 nm. Since a similar parameter $\Delta {{d}_{{\operatorname{tg} }}} = - {{d}_{{\operatorname{tg} }}}\delta {{C}_{{\operatorname{tg} }}}{\text{/}}{{C}_{{\operatorname{tg} }}}$ determined from the capacitance jump δC_tg does not exceed 0.7 nm, it is reasonable to attribute the jump of C_bg in Fig. 1b to the formation of the second electron layer near the bottom of the quantum well so that the distance between the layers is $d \sim \Delta {{d}_{{{\text{bg}}}}} \approx 25$ nm. This interpretation is confirmed by changes in the magnetocapacitance in low magnetic fields shown in Figs. 2a and 2b. Indeed, according to calculations for the model double quantum well and the experimental data obtained in [5, 12], the normalized magnetocapacitances ${{C}_{{{\text{bg}},n0}}}$ = ${{C}_{{{\text{bg}}}}}(B){\text{/}}{{C}_{{{\text{bg}}}}}(B = 0)$ and ${{C}_{{\operatorname{tg} ,n0}}}$ = ${{C}_{{\operatorname{tg} }}}(B){\text{/}}{{C}_{{\operatorname{tg} }}}(B = 0)$ between the electron system and different gates in the absence of resistive effects should coincide with each other in the case of the existence of only one layer, as observed in Fig. 2a measured at V_bg = 0.2 V < V_th. The capacitance curves ${{C}_{{\operatorname{tg} ,n0}}}$ and ${{C}_{{{\text{bg}},n0}}}$ at V_bg = 1.0 V > V_th (see Fig. 2b) in low magnetic fields demonstrate magnetic oscillations with different periods, which can be used to determine the electron density in the layer nearest to the corresponding gate [9]. Such studies of similar samples with the 60‑nm‑wide quantum well [9] showed that the main consequence of the increase in V_bg in the bilayer state (V_bg > V_th, low magnetic field) is an increase in the electron density in the bottom layer. A small decrease in the electron density in the top layer is simultaneously observed. This behavior of the electron density in bilayer systems is well known (see, e.g., [13] for the case of double quantum wells) and is due to the negative contribution to the compressibility of electrons, which is larger in absolute value in the layer with a lower electron density.

Fig. 2.

Normalized magnetocapacitances (black lines) ${{C}_{{{\text{bg}},n0}}}$ = ${{C}_{{{\text{bg}}}}}(B){\text{/}}{{C}_{{{\text{bg}}}}}(B = 0)$ and (gray lines) ${{C}_{{\operatorname{tg} ,n0}}}$ = ${{C}_{{\operatorname{tg} }}}(B){\text{/}}{{C}_{{\operatorname{tg} }}}(B = 0)$ in the case of the filling of (a) one (V_bg < V_th) and (b) two (V_bg > V_th) layers of electrons. The black line ${{C}_{{{\text{bg}},n0}}}$ in panel (b) is shifted downward by 0.03 for clarity. Minima on the capacitance curves marked by triangles correspond to indicated filling factors in (ν, empty symbols in panel (a)) the presence of the top layer only, (ν_TL, gray filled symbols in panel (b)) top, and (ν_BL, black filled symbols in panel (b)) bottom layers of the bilayer electron system.

The main result of this work is the observation of the transition of the studied system at V_bg > V_th from the bilayer state in low magnetic fields to the single-layer state in high fields, where the single remaining electron layer is localized near the top of the quantum well, as in the single-layer system at V_bg < V_th. This transition is indicated by the measurements of the magnetocapacitance C_bg presented in Fig. 3 as a function of the parameter ν⁻¹ inverse to the total filling factor ν = n_t/N₀, where N₀ = eB/(hc) is the degeneracy of one spin sublevel. The inverse total filling factor ν⁻¹ = eB/(hcn_t) is proportional to the magnetic field B with the coefficient of proportionality determined by the total electron density n_t. Figure 3 presents 4 of 14 curves from one series of measurements at V_tg = 0 and various V_bg values in the range of 0‒2.6 V. A key experimental result in this figure is the coincidence of the capacitances measured at all V_bg (and ${{n}_{{\text{t}}}}$) values that is observed near ν⁻¹ = 2/3 (ν = 3/2), although these capacitances at ν⁻¹ = 0 are different for V_bg < V_th and V_bg > V_th. This fact was established at all 14 V_bg values in the studied range, which is confirmed by the experimental points shown in Fig. 1b. This independence of the capacitance C_bg measured at ν = 3/2 from V_bg implies that the electron system at the corresponding filling factor is a single layer located as at V_bg = 0, i.e., near the top of the quantum well. To confirm this hypothesis and to determine the magnetic field region of existence of the single-layer state, we compared the magnetocapacitance curves ${{C}_{{{\text{bg}},n}}}(B)$ and ${{C}_{{{\text{fg}},n}}}(B)$ normalized to the capacitance values at ν = 3/2: ${{C}_{{j,n}}}(B)$ = ${{C}_{j}}(B){\text{/}}{{C}_{j}}(\nu = {\text{3/2}})$, where $j$ = bg or tg. The corresponding curves are presented in Fig. 4. It is seen that the normalized capacitances measured with different gates in high magnetic fields almost coincide with each other at any gate voltage. This coincidence is observed beginning with the magnetic field approximately corresponding to the filling factor ν = 2 for all studied bottom gate voltages V_bg > V_th, including two values presented in Fig. 4. Thus, we conclude that the formation of the state of the quantum Hall effect with the total filling factor ν = 2 in the studied samples is accompanied by the transition of the electron system to the single-layer state, which holds in higher magnetic fields at lower filling factors, in particular, in compressible states at $2 > \nu > 1$. In addition, we note the coincidence of the amplitudes of the dips in the capacitances C_{bg, n} and C_{tg, n} in Fig. 4a at filling factors ν = 4/3 and 5/3, which correspond to the states of the fractional quantum Hall effect. The observation of two such states is additional evidence of the single-layer character of the electron system because it is natural for the single-layer system, whereas the explanation of the state with ν = 5/3 in the bilayer system would require special assumptions. At low temperatures, resistive effects [14] inevitably influence the magnitude of deep minima in the capacitance (these effects are certainly significant in our case if the dips in the capacitance in states of the quantum Hall effect exceed 10%). Curves plotted in Fig. 4b were measured at T = 3 K, when resistive effects were absent in the entire magnetic field range. Figure 4b demonstrates very good coincidence of the amplitudes of dips in different normalized capacitances at filling factors of 1 and 2, proving the single-layer character of the system in the corresponding states of the quantum Hall effect, as observed in [5].

Fig. 3.

Four representative magnetocapacitance curves versus the inverse filling factor ν⁻¹ = eB/hcn_t, which is proportional to the magnetic field, measured at V_tg = 0 and different gate voltages V_bg indicated in the figure (and, correspondingly, at different electron densities n_t). The main result seen in this figure is the independence of the capacitance ${{C}_{{{\text{bg}}}}}({{\nu }^{{ - 1}}} \approx {\text{2/3}})$ measured at a filling factor of ν = 3/2 of the bottom gate voltage V_bg despite the capacitance jump in zero magnetic field (cf. the gray solid line with the three other lines, also see Fig. 1).

Fig. 4.

Normalized magnetocapacitances (black lines) ${{C}_{{{\text{bg}},n}}}(B)$ = ${{C}_{{{\text{bg}}}}}(B){\text{/}}{{C}_{{{\text{bg}}}}}(\nu = 3{\text{/}}2)$ and (gray lines) ${{C}_{{\operatorname{tg} ,n}}}(B)$ = ${{C}_{{\operatorname{tg} }}}(B){\text{/}}{{C}_{{\operatorname{tg} }}}(\nu = 3{\text{/}}2)$ measured at V_tg = 0 and (a) V_bg = 2.6 V, T = 0.5 K and (b) V_bg = 2.2 V, T = 3 K. Triangles mark the filling factors ν discussed in the main text. The horizontal square brackets labeled with SL indicate the regions of existence of the single-layer state of the electron system.

Further, we discuss the coincidence of the capacitances measured at different electron densities at the filling factor ν = 3/2 and their divergence at different filling factors observed in Fig. 3. This property is due to the strong dependence of the compressibility of two-dimensional electron systems in the quantizing magnetic field on the filling factor and the field intensity. Points with the same abscissa on different curves correspond to different magnetic fields. In particular, minima of the compressibility corresponding to states of fractional quantum Hall effect at filling factors ν = 5/3 and 4/3 become pronounced in the highest magnetic field (on the black solid line). At the filling factor ν = 3/2, no incompressible states are expected, and the Fermi level is located in the center of the upper spin sublevel, where the density of single-particle states and, correspondingly, its contribution to the compressibility of the two-dimensional electron system are high. In [15], we established that a change in the compressibility of two-dimensional electron systems with increasing magnetic field from zero to the value corresponding to the filling factor ν = 1/2, when the Fermi level is in the center of the lower spin sublevel, is close to a change in the compressibility of a system of noninteracting electrons. This relation indicates that the contribution from effects of interaction to the compressibility of the two-dimensional electron system is approximately the same in zero field and in the magnetic field corresponding to the filling factor ν = 1/2. As a result, the capacitances C of the field-effect transistor in these two states are related as [15]

$${{C}^{{ - 1}}}(\nu = {\text{1/2}}) = {{C}^{{ - 1}}}(B = 0) - \pi {{\hbar }^{2}}{\text{/}}m\text{*}{\kern 1pt} {{e}^{2}}S.$$

(1)

Here, $m\text{*}$ is the effective mass of two-dimensional electrons and $S$ is the area of the two-dimensional system under the gate. In our case, this relation is also valid for the filling factor ν = 3/2 because the single-particle density of states at this filling factor is high. This is demonstrated in Fig. 1b, where the capacitance of the electron system at ν = 3/2 measured at V_bg = 0, i.e., in the presence of only the top layer, is compared to the calculated value.

Electronic states of the bilayer electron system can be characterized not only by their distribution between the layers, as above in this work, but also by their belonging to one of two size-quantized subbands. The corresponding classes of states coincide in the case of a weak tunnel coupling between the layers and a significant difference between the electron densities in the layers (asymmetry of the electron system). In the latter case, electrons of two different subbands are localized in different layers. The electron distribution drastically changes in the symmetric (balanced) state in the presence of tunneling. In this case, the electron densities are identical in different layers and are different in different subbands corresponding to symmetric and antisymmetric wavefunctions. The energy difference between the bottoms of the subbands (intersubband splitting) $\Delta$ reaches the minimum value Δ_SAS, which is a measure of the tunnel coupling between the layers and is widely known as symmetric−antisymmetric splitting. The intersubband splitting can be determined from the positions of maxima of the intersubband magnetoresistance, which are determined by the condition of the coincidence of Landau levels in two different subbands: Δ = n$\hbar$ω_c, where $n = 1,2,3,...$ and ω_c is the cyclotron frequency (see, e.g., [16] and references therein). This condition is equivalent to the commensurability of the difference between electron densities in two subbands Δn_s with the degeneracy of the Landau level: Δn_s = 2nN₀. Measured Δn_s(V_bg) values are presented in Fig. 5. The minimum in this dependence corresponds to the balance point at V_bg = V_bal ≈ 2 V, when the electron density in the top layer n_s, TL is equal to the electron density in the bottom layer n_{s, BL}. The key result that the capacitance C_bg at ν = 3/2 is independent of the bottom gate voltage V_bg is valid for all studied V_bg values both at V_bg < V_bal and at V_bg > V_bal (see Figs. 1b and 3). Consequently, electrons in the single-layer state were localized near the top of the quantum well in our experiment at any relation between the electron densities in different layers of the bilayer electron state both at n_s, TL > n_s, BL and at n_s, TL < n_{s, BL}.

Fig. 5.

Difference of the electron densities Δn_s in two quantum-confined subbands determined from the period of the intersubband magnetoresistance oscillations at V_tg = 0 and T = 4.2 K versus V_bg. The position of the minimum of Δn_s(V_bg) at V_bg = V_bal corresponds to the electron configuration with the same electron density in different layers. The top scale shows the total electron density in the system n_t.

We note that the observed transition from the bilayer to single-layer state corresponds to the transition of electrons from the minimum of the potential near the bottom of the quantum well to the minimum near the top of the quantum well, as schematically shown in Figs. 6b and 6c. In this case, the deep minimum of the potential near the bottom of the quantum well that is empty of electrons seems surprising (Fig. 6c). However, this fact can be qualitatively explained by the negative contribution to the compressibility of two-dimensional electron systems from the exchange interaction. This contribution was determined theoretically [17] and was directly measured experimentally [18, 19]. A decrease in the electron density leads to an increase in the absolute value of this contribution, which is accompanied by an increase in the chemical potential, promoting the depletion of the layer with the lower density. Furthermore, the idea that the exchange interaction between electrons can result in the spontaneous filling of only one of two nearby quantum wells was proposed in [20] for case of zero magnetic field and initiated extensive discussions [21–24]. The implementation of this idea is complicated by an increase in the kinetic energy of electrons and the energy of the electric field at the filling of only one of two quantum wells. In our case of the high magnetic field (ν < 2) and narrow Landau levels, the transition of electrons between the wells is not accompanied by change in their kinetic energy equal to half of the cyclotron energy. This factor is apparently decisive for the implementation of the transition observed in our experiment just in the high magnetic field. In addition, the compressibility of the electron system in the case of narrow Landau levels is completely determined by the electron−electron interaction and is negative (see, e.g., [18, 19]). A change in the energy of the electric field in our asymmetric structure with doping only on one side of the quantum well promotes the observed transition of electrons to the top layer nearest to the donor layer. This can be qualitatively explained as follows. If the top gate, which is kept at V_tg = 0, is neglected for simplicity, the electric field above the quantum well is determined by the charge density of ionized donors, and is independent of the electron distribution across the quantum well. At the same time, the electric field on the other side of the quantum well can vary because the bottom gate voltage is fixed. In this case, the transition of electrons from the bottom to top layer is accompanied by a decrease in the capacitance of the parallel plate capacitor formed by the bottom gate and the electron system, which, at a constant gate voltage, corresponds to a decrease in the energy of the electric field. Thus, the observed redistribution of electrons can be qualitatively explained by a high degree of degeneracy of Landau levels, the negative compressibility of electrons, and a special asymmetric architecture of the used field-effect transistor. The future complete description of the observed effect requires the self-consistent calculation of size-quantized subbands in the wide quantum well including exchange and correlation effects.

Fig. 6.

Schematic shape of the potential well and the electron density distribution across quantum well in the top, |Ψ_t|², and bottom, |Ψ_b|², layers under various experimental conditions. (a) Single-layer electron system at V_bg = 0 in the presence of the top layer only and B = 0. (b) Bilayer electron system in the symmetric configuration at V_bg = V_bal and B = 0. (c) Single-layer electron system at V_bg = V_bal in high magnetic fields at ν < 2.

The results of a similar calculation for states of the quantum Hall effect in the wide quantum well at the filling factors ν = 2 and 4 in the local density approximation were briefly reported in [7]. The calculation was performed to explain experimental data on the hysteresis of the magnetoresistance. As a result, the observed hysteresis was attributed to the first-order quantum transitions resulting in the redistribution of electrons between the layers, which is accompanied by the complete depletion of one of the Landau levels. In the case of the symmetric quantum well, the redistribution of electrons leads to the spontaneous symmetry breaking with the possibility of opposite signs of the asymmetry parameter in different lateral regions of the sample, which corresponds to the appearance of a domain structure. Similar to our result, in the case of a filling factor of 2, such redistribution would mean the complete depletion of one of the layers, which was not noticed in [7]. We recall that all results obtained in [7] were obtained for the integer filling factors ν = 2 and 4.

To summarize, we have observed the transition of the electron system, which is formed in a wide GaAs quantum well, from the bilayer state in zero magnetic field to the single-layer state in the quantizing magnetic field at a filling factor of 2 for spin-resolved Landau levels. This transition corresponds to the filling of the two-dimensional layer located near the wall of the quantum well nearest to the selectively doped layer at any relation between the electron densities in different layers in zero magnetic field. A qualitative explanation of the observed effect has been proposed.