The conditions for the appearance of quantum Hall effect (QHE) states in bilayer electron systems (ESs) formed in double [1] and wide [2] GaAs quantum wells, as well as in graphene bilayers [3, 4], are significantly different from those in single-layer two-dimensional ESs. Effects of the electron–electron interaction, including correlation between electrons in different layers almost always play a significant role in the formation of QHE states in bilayer systems. As a result, fractional QHE (FQHE) states with total filling factors νF of spin-resolved Landau levels with conventional odd and even denominators such as νF = 1/2 [5, 6] and νF = 1/4 [7, 8] are observed in such systems. Furthermore, they also demonstrate an unconventional integer QHE state with a filling factor of 1 with unique transport characteristics related to the formation of an exciton condensate (see review [9] for double GaAs quantum wells and recent works [10, 11] for graphene bilayer).

Another feature of QHE states in bilayer ESs is a complex behavior in a magnetic field tilted from the normal to the plane of the system when the filling factors of the observed states can significantly vary. Such an effect was clearly manifested in balanced (having the same electron density) bilayers formed in wide GaAs quantum wells [12, 13]. New FQHE states observed in those works in the tilted magnetic field were attributed [12] to the spontaneous breaking of balance because of the redistribution of electrons between the layers and the formation of combined FQHE states from appropriate states of individual layers. We note that the FQHE state with a total filling factor of 1/4 was also observed for the first time in a sample with a wide quantum well in the tilted magnetic field [7].

The aim of this work is to study strongly imbalanced bilayer ESs, where the nearest neighbors of electrons of the lower density layer are electrons of the other layer and the interlayer Coulomb interaction for these electrons can thereby be stronger than the intralayer interaction. This condition is impossible in a balanced system with a finite distance between the layers because the distances between electrons in both layers are identical.

We study the effect of the tilted magnetic field on QHE states in the imbalanced bilayer ES that is formed in a 60-nm-wide GaAs quantum well and has the ratio of the electron densities in different layers in zero magnetic field exceeding 2.4. It is found that QHE states with the total filling factors ν = 1 and 2, which dominate in the normal magnetic field, disappear in the magnetic field tilted at the angle Θ = 30°–50° depending on the filling factor and the ratio of the electron densities in different layers. Instead, FQHE states appear at filling factors νF = 4/3, 10/7, 6/5, and 5/4 corresponding to the quantum numbers q = 3/4, 7/10, 5/6, and 4/5 in the Hall resistance \({{R}_{{xy}}} = qh{\text{/}}{{e}^{2}}\). The detection of the FQHE state with a quantum number of 4/5 is the most remarkable because the indication of its existence was obtained to date only in measurements of the dissipative conductivity in balanced graphene bilayers in the Corbino geometry [3]. The combination of magnetotransport and magnetocapacitance measurements in this work makes it possible to characterize the distribution of electrons between the layers and to reveal a combined character of the observed FQHE states. The results obtained indicate a change in the character of electron correlations in the tilted field and the redistribution of electrons between the layers under the variation of the field. This redistribution results in the formation of combined FQHE states with filling factors νF corresponding to the filling factor 1 in the layer with the higher electron density (νul = 1) and the filling factor νdl = νF – 1 in the lower density layer. To the best of our knowledge, this is the first observation of various combined FQHE states corresponding to the same filling factor in one of the layers under the variation of the magnetic field. The detection of the 5/4 state indicates the effect of the layer with the filling factor 1 on the intralayer electron–electron interaction in the neighboring layer.

Experiments were carried out with two samples prepared from one GaAs/AlGaAs heterostructure with the 60‑nm-wide GaAs quantum well located at a depth of 140 nm from the heterostructure surface. Identical results were obtained for both samples. Selective doping, which ensured the filling of the well with electrons, was performed above the well at a distance of 65 nm from its upper edge and led to a strong initial asymmetry of the well. Field electrodes (gates) were created on both sides of the well. The back gate was a heavily doped GaAs layer formed during the growth of the heterostructure and was located at a distance of 850 nm below the lower edge of the well. The front gate was formed by depositing a thin metal film on the heterostructure surface. A two-dimensional electron system (below, the upper layer) with a density of nu = 1.8 × 1011 cm–2 and a mobility of μu = 1.1 × 107 cm2/(V s) arose near the upper edge of the quantum well at zero gate voltage. To reach filling factors of interest in the tilted magnetic field, most measurements were carried out at the front gate voltage Vug = –0.2 V. At this voltage, the second two-dimensional layer of the ESs (below, lower layer) appeared near the lower edge of the quantum well at the back gate voltage Vdg > +0.15 V and the total electron density nt > 1.65 × 1011 cm–2. Leakages to the gates and the instability of the electron density limited the range of variation of the total density nt in the bilayer samples under study from 1.4 × 1011 to 2.4 × 1011 cm–2. All measurements were performed in this range, where the specified artifacts were absent. Samples had the lateral shape of Hall bars. Ohmic contacts to both layers were created by the diffusion of the NiGeAu alloy. The used electric circuit allowed simultaneous measurement of the magnetoresistance and Hall resistance of a sample, as well as electric capacitances between both gates and the electron system. The samples under study and the measurement circuit were described in detail in [14]. The magnetic field dependences of capacitances in low magnetic fields had minima at integer filling factors of Landau levels with electrons of the layer nearest to a given gate. This made it possible to directly determine the electron densities in each layer [14] and to detect their change under the variation of the magnetic field. The electron densities in the lower layer nd determined from the measured capacitance between the back gate and the ES in low magnetic fields are given in Figs. 1–4 together with the total electron density nt determined from the positions of pronounced integer QHE states. Minima of both capacitances observed simultaneously in a high perpendicular magnetic field at filling factors of 1 and 2 indicated the formation of a single-layer ES in the corresponding QHE states [15]. The sample was mounted on a rotatable platform ensuring the smooth rotation of the quantum well plane with respect to the magnetic field of a superconducting magnet with a critical magnetic field of 12 T. The rotation angle was calibrated with an accuracy of 2°. A higher accuracy of the determination of the rotation angle of about 0.2° could be obtained by comparing the Hall resistances in perpendicular and tilted magnetic fields. The sample was placed in liquid 3He. The measurements were carried out at a temperature of 0.5 K obtained by the evacuation of 3He vapor.

Fig. 1.
figure 1

Magnetoresistance Rxx (right scale) and Hall resistance Rxy (left scale) versus the normal magnetic field component Bn for two angles between the field and the normal to the quantum well plane Θ = (gray solid lines) 0° and (black solid lines) 48°. The gray Hall resistance line is shifted downward by 0.05 for clarity. Vertical dashed and dotted lines mark filling factors ν and νF that are discussed in the main text and are determined from the position of the ν = 3 integer quantum Hall effect state. Horizontal straight segments indicate quantized Hall resistances for fractional quantum Hall effect states with quantum numbers 3/4 and 4/5. The electron densities in the lower layer nd in weak magnetic fields and the total electron density in the system nt are given in units of 1010 cm–2.

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Fig. 2.
figure 2

Hall resistance Rxy versus the total magnetic field at different tilt angles Θ. The leftmost and next lines were measured at Θ = 0° and 15°, respectively, and all other lines were measured with a step of ΔΘ = 5°. The horizontal dotted lines mark quantized Hall resistances. Two gray lines demonstrate the disappearance of quantum Hall effect states with Rxy = h/e2 and 1/2(h/e2).

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Fig. 3.
figure 3

Hall resistance Rxy (left scale), magnetoresistance Rxx (right scale), and variation \(\Delta C\) of the capacitance between the front gate and the electron system versus the inverse total filling factor ν–1 = eBn/(hcnt) proportional to the normal component of the magnetic field at the electron densities indicated in the legends. All data in each panel were measured simultaneously, under the variation of the magnetic field tilted by the angle Θ = 40°, and at a temperature of T = 0.5 K. Vertical straight lines mark the fractional filling factors νF = 10/7, 4/3, and 5/4. The total widths of minima in capacitance are indicated by horizontal straight segments and positions of extrema are indicated by triangles.

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Fig. 4.
figure 4

Same as in Fig. 3, but measured at much lower electron densities nd at which the νF = 6/5 fractional quantum Hall effect state is observed and the νF = 4/3 state can be absent (panel (a)).

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Figure 1 shows magnetotransport curves (magnetoresistance Rxx and Hall resistance Rxy) measured in perpendicular and tilted magnetic fields. First, the filling factor-2 QHE state disappears completely in the tilted field. It is demonstrated below that the same is true for the filling factor-1 state. The suppression of the QHE in the tilted magnetic field at ν = 1 and 2 was observed previously in bilayer electron systems formed in both double and wide quantum wells. However, this suppression strongly depends on various conditions, e.g., on the balance in the bilayer electron system. In particular, the suppression of the quantum Hall effect at a filling factor of 2 in samples with double quantum wells was observed in [16] only in the case of the imbalanced system. The authors of [12, 13] observed only the disappearance of the ν = 1 QHE state in a wide quantum well. Here, we only mention the identical behavior of the ν = 1 and 2 QHE states in the tilted magnetic field; the main result of this work is the detection of unconventional FQHE states in such a field. Figure 1 demonstrates two FQHE states mostly corresponding to the filling factors νF = 4/3 and 5/4 in the tilted field. The latter state has not yet been observed in GaAs-based semiconductor heterostructures. The authors of [12, 13] reported the appearance of FQHE states with filling factors with both even (νF = 4/5 and 6/5) and odd (νF = 11/15 and 19/15) numerators in the initially balanced bilayer ES in the wide GaAs quantum well in the tilted field. These states appeared in addition to states with filling factors νF = 2/3 and 4/3, which also existed in the perpendicular field. All these states were interpreted as combined FQHE states in different layers. States with even numerators can be implemented in the balanced ES. To explain states with odd numerators, it was assumed that they are due to spontaneous balance breaking accompanied by the redistribution of electrons between the layers. In particular, the state with the filling factor νF = 11/15 (νF = 19/15) was attributed to the combination of the νF1 = 1/3 and νF2 = 2/5 (νF1 = 2/3 and νF2 = 3/5) states in layers 1 and 2, respectively. The fraction 19/15 differs from 5/4 by 1.3%. The filling factors νF = 19/15 and 6/5 are also marked in Fig. 1. They describe worse the position of the right Rxx minimum of the pair, which arose in the tilted field. For this reason, we associate this minimum with νF = 5/4. A possible nature of FQHE states observed in this work will be discussed below.

The evolution of QHE states in the tilted magnetic field is presented in Fig. 2, where the Hall resistance measured at different Θ angles is shown as a function of the total magnetic field. In this representation, the average slope of Hall curves is proportional to cosΘ. Pronounced horizontal sections of the curves correspond to quantized Hall resistance. The quantum plateaus with Rxy = h/e2 and 1/2(h/e2) in the presented data disappear at Θ ≈ 30° and 40°, respectively. At the same time, a Hall plateau with Rxy = 3/4(h/e2) becomes pronounced at Θ > 25° and a plateau appears with the Hall resistance tending to Rxy = 4/5(h/e2) with increasing Θ > 30°.

The variation of the electron density in the ES and its initial distribution between the layers changes the observed FQHE states. In particular, in addition to states with the filling factors νF = 4/3 and 5/4 presented in Figs. 1 and 2, we observed FQHE states with the filling factors νF = 10/7 and 6/5 (see Figs. 3 and 4).

We established the condition of appearance of certain states by analyzing the measured capacitances between the gates and the ES. These measurements showed that all observed FQHE states occur simultaneously with the broad minimum in the capacitance between the front gate and the electron system corresponding to a filling factor of 1 in the upper layer with the higher electron density (minima in the capacitance between the gates and the bilayer ES were discussed in [14, 15], see also Fig. 5). This result is illustrated by the data in Figs. 3 and 4 obtained for each figure simultaneously under the same variation of the magnetic field. The main criteria of the appearance of a certain FQHE state is its coexistence with the capacitance minimum and proximity to the center of this minimum marked by the triangle in Figs. 3 and 4. In particular, the leftward displacement of the triangle in Fig. 3 leads to the disappearance of the right minimum in the magnetoresistance (νF = 5/4) and to the appearance of the pronounced left minimum (νF = 10/7) while maintaining the dominant central minimum (νF = 4/3) closest to the triangle. The data in Fig. 4 confirm that the formulated criteria are valid. The state with νF = 4/3 dominant in Fig. 3 is not observed in Fig. 4 when it is beyond the minimum (Fig. 4a), and it again becomes dominant (in this case, compared to the νF = 6/5 state present in both Figs. 4a and 4b) when it falls in the corresponding interval (Fig. 4b).

Fig. 5.
figure 5

Capacitance Cug between the front gate and the electron system versus the normal magnetic field in the cases of filling of (gray line) only one layer and (black line) two layers. Deep minima in the capacitance correspond to integer filling factors of Landau levels with all electrons of the system. They are marked by dashed vertical lines with the total filling factors in the single-layer (ν1) and two-layer (ν) states. The horizontal segments specified by νul = 1 and νul = 2 show the total widths of the minima in the capacitance corresponding to these filling factors νul in the upper layer of the electron system. The triangle marks the minimum with νul = 1 corresponding to minima in Figs. 3 and 4. The width of this minimum is denoted as \({{\Delta }^{{(2l)}}}\). For comparison, the gray horizontal segment presents the total width \({{\Delta }^{{(1l)}}}\) of the minimum in the capacitance in the single-layer state with \({{\nu }_{1}} = 1\).

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We now discuss a reason for the established correlation between the positions of minima in the capacitance and FQHE states. It is well known that the chemical potential of the two-dimensional electron system in increasing magnetic field is shifted to a lower Landau level or a spin sublevel at an integer filling factor. This corresponds to the minimum in the compressibility of the electron system which is manifested in the minimum of the quantum correction to the capacitance of a field-effect transistor with the two-dimensional electron system [17, 18]. An additional (and often leading) contribution to the amplitude of the minimum in the measured capacitance can come from resistive effects caused by long charging times of the two-dimensional ES in a QHE state because of a small value of its dissipative conductivity [19]. In the case of two layers with different electron densities in equilibrium, such a jump of the chemical potential occurring in one of the layers results in the transfer of electrons from the neighboring layer to this layer, maintaining an integer filling factor in it. This effect leads to a significant broadening of the corresponding incompressible states in an individual layer in the magnetic field, which was observed in samples with double quantum wells in [20–22] and was explained in [22] as described above. A similar effect is also present in samples with the bilayer system in the wide quantum well studied here, as is demonstrated in Fig. 5. This figure presents the capacitance curves measured between the front gate and the ES at the filling of only the upper layer (gray line) and at the filling of two layers (black line). The black line has two types of minima. Deep minima correspond to the filling factors ν = 1 and 2 of spin-resolved Landau levels for all electrons of the bilayer system (total filling factors), whereas shallower minima correspond to the same filling factors νul = 1 and 2 for electrons of the upper layer [15, 22]. Both curves were measured at almost identical initial electron density in the upper layer, as is obvious from the coincidence of the positions of minima in the capacitance corresponding to the filling factor ν1 = 2 in the single-layer system and the filling factor νul = 2 in the upper layer of the bilayer system. Since the single-layer system is formed by the upper layer of electrons, the only qualitative difference between states with the same ν1 and νul is the existence of the second (lower) layer of electrons in the latter case. A small depth of minima in the capacitance at integer filling factors νul is due to the screening of the electric field of the gate by the second layer. We also note that, according to the experiments, resistive effects make a noticeable contribution to the depth of deep minima in the capacitance and are absent in the region of shallow minima. A key circumstance for our analysis is a significant difference between the total widths of deep and shallow minima corresponding to the same filling factors ν1 and νul. In particular, \({{\Delta }^{{(2l)}}}{\text{/}}{{\Delta }^{{(1l)}}} > 2\) for data in Fig. 5. The minima at the total filling factors ν are narrower than minima at the same filling factors ν1. Shallow minima have a large width just because an integer filling factor in the upper layer under the variation of the magnetic field is maintained by the transfer of electrons from the lower layer. Thus, the QHE in this case effectively occurs in a system with an electron reservoir [23, 24], which is the lower layer with the lower electron density. The relative change in the electron density in the upper layer within the minimum region with νul = 1 can be estimated from the relation \(\Delta {{n}_{u}}{\text{/}}{{n}_{u}} \approx ({{\Delta }^{{(2l)}}} - {{\Delta }^{{(1l)}}}){\text{/}}\tilde {B}\), where \(\tilde {B}\) is the magnetic field in the center of the minimum. For the data in Fig. 5, this estimate is about 10%. Since the electron density in the lower layer is noticeably lower, the relative decrease in the electron density in it is much larger; as a result, different fractional filling factors in the lower layer can be reached in the region of one minimum in the capacitance, as seen in Figs. 3 and 4.

Consequently, unconventional FQHE states observed in this work are naturally interpreted as combined QHE states with a filling factor of 1 in the upper layer and a filling factor of νF – 1 in the lower layer. Thus, FQHE states with filling factors of 1/3, 1/5, and 3/7 with odd denominators and with a filling factor of 1/4 in the lower density layer occur in our case. Within the proposed explanation, it is easy to calculate a change \(\Delta n\) in the electron density in the layers under the variation of the magnetic field between two FQHE states with filling factors νF1 and νF2 (νF1 < νF2). This change in the upper layer is given by the formula

$$\Delta {{n}_{{\text{u}}}} = e\left( {{{B}_{{\text{n}}}}({{\nu }_{{{\text{F}}1}}}) - {{B}_{{\text{n}}}}({{\nu }_{{{\text{F}}2}}})} \right){\text{/}}hc = {{n}_{{\text{t}}}}(\nu _{{{\text{F}}1}}^{{ - 1}} - \nu _{{{\text{F}}2}}^{{ - 1}}),$$
(1)

where nt is the total electron density in the ES. For the lower layer, Δnl = –Δnu. In particular, for the data in Fig. 3, Δnu = 0.05nt for the transition between the states with νF2 = 10/7 and νF1 = 4/3 or νF2 = 4/3 and νF1 = 5/4. Thus, the above estimate for a change in the electron density in the layers from the broadening of the minimum in the capacitance is only slightly smaller than the value necessary for the appearance of the three mentioned FQHE states and is sufficient for the observation of pairs of states including the state with νF = 4/3. On the other hand, it is easy to estimate that the implementation of the combined state with the filling factor νF = 19/15 = 2/3 + 3/5 at the transition from the state with νF = 4/3 = 1 + 1/3 would require much larger redistribution of electrons between the layers corresponding to \(\Delta {{n}_{{\text{u}}}} = - (17{\text{/}}76){{n}_{{\text{t}}}}\) ≈ \( - 0.22{{n}_{{\text{t}}}}\), which seems hardly possible because of much smaller jumps of the chemical potential in FQHE states with filling factors of 2/3 and 3/5 compared to a filling factor of 1. This estimate indeed confirms the detection of the combined FQHE state with the filling factor νF = 5/4 = 1 + 1/4. For completeness, we also note an increase in the electron density in the upper layer at the filling factor νul = 2 (see Fig. 5), which rightward shifts the center of the νul = 1 minimum with respect to the ν1 = 1 line.

The existence of FQHE states with filling factors with odd denominators is well established for single-layer ESs; the appearance of the state with a filling factor of 1/4 in the lower density layer should be due to some specific features. We believe that this state in our case appears because the interaction of electrons of the lower layer with electrons of the upper layer is stronger than the intralayer interaction. In particular, the electron density in the lower (upper) layer in the combined FQHE state with the filling factor νF = 5/4 is ñd = nt/5 (ñu = 4nt/5). Then, for the data presented in Fig. 1, the average distance between electrons in the lower layer is ad–d = 2(π ñd)–1/2 > 55 nm, whereas the average distance between electrons of the lower and upper layers is \({{a}_{{{\text{d}} - {\text{u}}}}} = \sqrt {{{d}^{2}} + 4(\pi {{{\tilde {n}}}_{{\text{u}}}}{{)}^{{ - 1}}}} \) < 44 nm. Here, d = 34 nm is the experimentally determined effective distance between the layers (see [14]).

We briefly discuss possible reasons for the appearance of observed FQHE states in the tilted field. Phenomenologically, the magnetic field component parallel to the layer plane significantly changes coupling between the layers in both the double quantum well with a fixed profile and the wide quantum well, where the bilayer system is formed in the self-consistent potential. In the former case, the parallel component of the magnetic field reduces the absolute values of the tunnel matrix elements coupling Landau levels of different layers (moreover, these matrix elements for nonzeroth Landau levels are oscillatory functions of the magnetic field) [25]. In the case of coherent interlayer states, the parallel component of the magnetic field additionally leads to the lateral change in the phase [26], which can form a lattice of solitons at a large magnitude of this component [27]. Numerical solutions [28] for the wide quantum well show that the parallel field increases effective distance between the layers, and experimental results obtained in [28] indicate a transition from the single-layer-like state of the ES in the perpendicular field to the bilayer state in the tilted field. It is worth noting that both bilayer and single-layer-like states, as well as magnetic-field-induced quantum phase transitions between such states, were detected in our work [15] in the nominally bilayer ES in the perpendicular magnetic field where they arise at different filling factors. We believe that the mentioned change in coupling between the layers caused by the magnetic field component parallel to the layers is responsible for the disappearance of QHE states with filling factors of 1 and 2 and for the appearance of observed FQHE states. The aim of future studies is to determine the role of each of the listed possible mechanisms.

To summarize, we have detected a drastic change in the picture of the quantum Hall effect caused by a tilted magnetic field has been detected in samples with a bilayer electron system in a wide quantum well integer quantum Hall effect states disappear, whereas fractional quantum Hall effect states with both odd and even denominators appear between the filling factors of 1 and 2 of spin-resolved Landau levels. It has been shown that the detected states are combinations of ν = 1 and fractional quantum Hall effect states in different layers, which are supported by the redistribution of electrons between the layers. We have indicated features of the inter- and intralayer Coulomb interaction in the studied samples, where the electron densities in different layers are very different.