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Tunable insulator-quantum Hall transition in a weakly interacting two-dimensional electron system


We have performed low-temperature measurements on a gated two-dimensional electron system in which electron–electron (e-e) interactions are insignificant. At low magnetic fields, disorder-driven movement of the crossing of longitudinal and Hall resistivities (ρxx and ρxy) can be observed. Interestingly, by applying different gate voltages, we demonstrate that such a crossing at ρxx ~ ρxy can occur at a magnetic field higher, lower, or equal to the temperature-independent point in ρxx which corresponds to the direct insulator-quantum Hall transition. We explicitly show that ρxx ~ ρxy occurs at the inverse of the classical Drude mobility 1/μD rather than the crossing field corresponding to the insulator-quantum Hall transition. Moreover, we show that the background magnetoresistance can affect the transport properties of our device significantly. Thus, we suggest that great care must be taken when calculating the renormalized mobility caused by e-e interactions.


At low temperatures (T), disorder and electron–electron (e-e) interactions may govern the transport properties of a two-dimensional electron system (2DES) in which electrons are confined in a layer of the nanoscale, leading to the appearance of new regimes of transport behavior [1]. In the presence of sufficiently strong disorder, a 2DES may behave as an insulator in the sense that its longitudinal resistivity (ρxx) decreases with increasing T[2]. It is useful to probe the intriguing features of this 2D insulating state by applying a magnetic field (B) perpendicular to the plane of a 2DES [24]. In particular, the direct transition from an insulator (I) to a high filling factor (v ≥ 3) quantum Hall (QH) state continues to attract a great deal of both experimental [513] and theoretical [1416] interest. This is motivated by the relevance of this transition to the zero-field metal-insulator transition [17] and by the insight it provides on the evolution of extended states at low magnetic fields. It has already been shown that the nature of the background disorder, in coexistence with e-e interactions, may influence the zero-field metallic behavior [18] and the QH plateau-plateau transitions [19, 20]. However, studies focused on the direct I-QH transitions in a 2DES with different kinds of disorder are still lacking. Previously, we have studied a 2DES containing self-assembled InAs quantum dots [11], providing a predominantly short-range character to the disorder. We observed multiple T-independent points in ρ xx (B), indicating a series of transitions between a low-field insulator and a QH state. The oscillatory amplitude of ρ xx (B) was well fitted by the Shubnikov-de Haas (SdH) theory [2123], with amplitude given by

Δ ρ xx B , T = C exp π / μ q B D B , T ,

where μq represents the quantum mobility, D(B, T) = 2π2kBm * T/eB sinh (2π2kBm * T/eB), and C is a constant relevant to the value of ρxx at B = 0 T. The observation of the SdH oscillations suggests the possible existence of a Fermi-liquid metal. It should be pointed out that the SdH theory is derived by considering Landau quantization in the metallic regime without taking localization effects into account [24, 25]. By observing the T-dependent Hall slope, however, the importance of e-e interactions in the metallic regime can be demonstrated [26]. In addition, as reported in [27], with a long-range scattering potential, SdH-type oscillations appear to span from the insulating to the QH-like regime when the e-e interaction correction is weak. Recently, the significance of percolation has been revealed both experimentally [28] and theoretically [29, 30]. Therefore, to fully understand the direct I-QH transition, further studies on e-e interactions in the presence of background disorder are required.

At low B, quantum corrections resulting from weak localization (WL) and e-e interactions determine the temperature and magnetic field dependences of the conductivity, and both can lead to insulating behavior. The contribution of e-e interactions can be extracted after the suppression of WL at B > Btr, where the transport magnetic field (Btr) is given by 4 eDτ with reduced Planck's constant (), electron charge (e), diffusion constant (D), and transport relaxation time (τ). In systems with short-range potential fluctuations, the theory of e-e interactions is well established [31]. It is derived based on the interference of electron waves that follow different paths, one that is scattered off an impurity and another that is scattered by the potential oscillations (Friedel oscillation) created by all remaining electrons. The underlying physics is strongly related to the return probability of a scattered electron. In the diffusion regime (kB/ < < 1 with Boltzmann constant kB), e-e interactions contribute only to the longitudinal conductivity (σxx) without modifying the Hall conductivity (σxy). On the other hand, in the ballistic regime (kB/ > > 1), e-e interactions contribute both to σxx and σxy, and effectively reduce to a renormalization of the transport mobility. However, the situation is different for long-range potential fluctuations, which are usually dominant in high-quality GaAs-based heterostructures in which the dopants are separated from the 2D electron gas by an undoped spacer. It is predicted that the interaction corrections can be suppressed at B = 0 but that they can eventually be restored at high magnetic fields B > 1/μD with enhanced return probability of scattered electrons, where μD represents the Drude mobility [32, 33]. Therefore, it is of great interest to study the direct insulator-quantum Hall transition in a system with long-range scattering, under which the e-e interactions can be sufficiently weak at low magnetic fields.

Theoretically, for either kind of background disorder, no specific feature of interaction correction is predicted in the intermediate regime where kB/ ≈ 1. Nevertheless, as generalized by Minkov et al. [34, 35], electron–electron interactions can still be decomposed into two parts. One, with properties similar to that in the diffusion regime, is termed the diffusion component, whereas the other, sharing common features with that in the ballistic limit, is known as the ballistic component. Therefore, by considering the renormalized transport mobility μ′ induced by the ballistic contribution and the diffusion correction δ σ xx d , σxx is expressed as

σ xx = neμ ' 1 + μ ' 2 B 2 + δ σ xx d ,
σ xy = neμ ' 2 B 1 + μ ' 2 B 2 .

It directly follows that the ballistic contribution δ σ xx b is given by δ σ xx b = ne μ ' μ D , where n is the electron density and μD is the transport mobility derived in the Drude model. After performing matrix inversion with the components given in Equations 2 and 3, the magnetoresistance ρxx(B) takes the parabolic form [36, 37]

ρ xx 1 neμ ' 1 neμ ' 2 1 μ ' 2 B 2 δ σ xx d

The Hall slope RH (ρxy/B with Hall resistivity ρxy) now becomes T-dependent which is ascribed to the diffusion correction δ σ xx d [38]. As will be shown later, Equations 3, 4, and 5 will be used to estimate the e-e interactions in our system. Moreover, both diffusive and ballistic parts will be studied.

As suggested by Huckestein [16], at the direct I-QH transition that is characterized by the approximately T-independent point in ρxx,

ρ xx , ρ xy

While Equation 5 holds true in some experiments [2], in others it has been found that ρxy can be significantly higher than ρxx near the direct I-QH transition [10, 28]. On the other hand, ρxy can also be lower than ρxx near the direct I-QH transition in some systems [39]. Therefore, it is interesting to explore if it is possible to tune the direct I-QH transition within the same system so as to study the validity of Equation 5. In the original work of Huckestein [16], e-e interactions were not considered. Therefore, it is highly desirable to study a weakly disordered system in which e-e interactions are insignificant. In this paper, we investigate the direct I-QH transition in the presence of a long-range scattering potential, which is exploited as a means to suppress e-e interactions. We are able to tune the direct I-QH transition so that the corresponding field for which Equation 5 is satisfied can be higher or lower than, or even equal, to the crossing field that corresponds to the direct I-QH transition. Interestingly, we show that the inverse Drude mobility 1/μD is approximately equal to the field where ρxx crosses ρxy, rather than the one responsible for the direct I-QH transition. We also show that the onset of strong localization occurs at a relatively higher field which does not correspond to 1/μD.


A gated modulation-doped AlGaAs/GaAs heterostructure (LM4640) is used in our study. The following layer sequence was grown on a semi-insulating GaAs substrate: 1 μm GaAs, 200 nm Al0.33Ga0.67As, 40 nm Si-doped Al0.33Ga0.67As with doping concentration in cubic centimeter, and finally a 10-nm GaAs cap layer. The sample was mesa etched into a standard Hall bar pattern, and a NiCr/Au gate was deposited on top of it by thermal evaporation. The length and width of the Hall bars are 640 and 80 μm, respectively. Four-terminal magnetotransport measurements were performed in a top-loading He3 system using standard ac phase-sensitive lock-in techniques over the temperature range 0.32 K ≤ T ≤16 K at three different gate voltages Vg = −0.125, −0.145, and −0.165 V.

Results and discussion

Figure 1a shows ρxx(B) and ρxy(B) at various T for Vg = −0.145 V. It can be seen from the inset in Figure 1 that the 2DES behaves as an insulator over the whole temperature range at all applied gate voltages. The Hall slope RH shows a weak T dependence below T = 4 K and is approximately constant at high T, which can be seen clearly in Figure 1b for each Vg. For 1.84 T < B < 2.85 T, a well-developed ν = 2 QH state manifests itself in the quantized ν = 2 Hall plateau and the associated vanishing of ρxx. In order to study the transition from an insulator to a QH state, detailed results of ρxx and ρxy at low T are shown in Figure 2a,b,c for each Vg, and the converted σxx and σxy are presented in Figure 3. At Vg = −0.125 V, spin splitting is resolved as the effective disorder is decreased compared to that at Vg = −0.145 and −0.165 V. The reason for this is that the carrier density at Vg = −0.125 V is higher than those at Vg = −0.145 and −0.165 V. Following the suppression of weak localization, with its sharp negative magnetoresistance (NMR) at low magnetic fields, the 2DES undergoes a direct I-QH at B = 0.26, 0.26, and 0.29 T ≡ Bc for Vg = −0.125, −0.145, and −0.165 V, respectively, since there is no signature of ν = 2 or ν = 1 QH state near Bc. We note that in all cases, Bc > 10 Btr. Therefore, it is believed that near the crossing field, weak localization effect is not significant in our system [37]. It is of fundamental interest to see in Figure 2d that the relative position of Bc with respect to that corresponding to the crossing of ρxx and ρxy is not necessarily equal. Following the transition, magneto-oscillations superimposed on the background of NMR are observed within the range 0.46 T ≤ B ≤ 1.03 T, 0.49 T ≤ B ≤ 1.12 T, and 0.53 T ≤ B ≤ 0.94 T for corresponding Vg, the oscillating amplitudes of which are all well fitted by Equation 1. The results are shown in Figure 4a,b,c for three different Vg. The good agreement with the SdH theory suggests that strong localization effects are not significant near Bc. This is consistent with our previous results, performed on both a delta-doped quantum well with additional modulation doping [13] and a modulation-doped AlGaAs/GaAs heterostructure with a superlattice structure [27]. It follows that we can obtain the quantum mobility μq from the fits, which is expected to be an essential quantity regarding Landau quantization. The estimated μq are 0.88, 0.84, and 0.77 m2/Vs for Vg = −0.125, −0.145, and −0.165 V, respectively. Moreover, from the oscillating period in 1/B, the carrier density n is shown to be T-independent such that a slight decrease in RH at low T does not result from the enhancement of carrier density n. Instead, these results can be ascribed to e-e interactions.

Figure 1
figure 1

Temperature dependence. (a) Longitudinal and Hall resistivities (ρxx and ρxy) as functions of magnetic field B at various temperatures T ranging from 0.3 to 16 K. The inset shows ρxx(B = 0, T) at three applied gate voltages. (b) Hall slope RH as a function of T at each Vg on a semi-logarithmic scale.

Figure 2
figure 2

Detailed results of ρ xx and ρ xy at low T . The B dependences of ρxx and ρxy at various T ranging from 0.3 to 1.5 K for (a) Vg = −0.125 V, (b) Vg =−0.145 V, and (c) Vg = −0.165 V. The insets are the zoom-ins of low-field ρxx(B). The dashed lines are the fits to Equation 4 at the lowest T. For comparison, the results at the lowest T for each Vg are re-plotted in (d). The T-independent points corresponding to the direct I-QH transition are indicated by vertical lines, and those for the crossings of ρxx and ρxy are denoted by arrows. Other T-independent points are indicated by circles.

Figure 3
figure 3

Converted σ xx ( B ) and σ xy ( B ) at various T ranging from 0.3 to 1.5 K. For (a) Vg = −0.125 V, (b) Vg = −0.145 V, and (c) Vg = −0.165 V. The insets show σxy(B) at T = 0.3 K and T = 16 K together with the fits to Equation 3 as indicated by the red lines. The vertical lines point out the crossings of σxx and σxy.

Figure 4
figure 4

ln ( Δρ xx ( B , T )/ D ( B , T )) as a function of 1/B . For (a) Vg = −0.125 V, (b) Vg = −0.145 V, and (c) Vg = −0.165 V. The dotted lines are the fits to Equation 1.

At first glance, the T-dependent RH, together with the parabolic MR in ρxx (denoted by the dashed lines in Figure 2 for each Vg), indicates that e-e interactions play an important role in our system. However, as will be shown later, the corrections provided by the diffusion and ballistic part of e-e interactions have opposite sign to each other, such that a cancelation of e-e interactions can be realized. Here we use two methods to analyze the contribution of e-e interactions. The first method is by fitting the measured ρxx to Equation 4, as shown by the blue symbols in Figure 5, from which we can obtain both δ σ xx b and δ σ xx d . The value of δ σ xx d is shown to be negative, as a result of the observed negative MR. We can see clearly from the dashed line in Figure 2 that the parabolic MR fits Equation 4 well at B > Bc but that it cannot be extended to the field where SdH oscillations occur. The obtained μ′, with an approximately linear dependence on T that is characteristic of the ballistic contribution of e-e interactions, is shown in Figure 6a,b,c for Vg = −0.125, −0.145, and −0.165 V, respectively. It should be mentioned that we cannot use this method to obtain μ′ for T > 4 K since there is no apparent parabolic NMR, as shown in Figure 1a. The second method is based on the analysis of σxy using Equation 3, as shown in the inset to Figure 3 at the highest and lowest measured T. In this approach, n is determined from the SdH oscillations, from which the renormalized mobility can also be obtained at high T even without the parabolic negative MR induced by the diffusion correction. Here we limit the fitting intervals below 0.75 Bmax to avoid the regime near μDB ~ 1, where Bmax denotes the field corresponding to the appearance of maximum σxy at the lowest T. The fitting results are plotted at each Vg as red symbols in Figure 6, allowing a comparison with those obtained by the first method. The figures show that μ′ is proportional to T when T > 4 K. There is a clear discrepancy between the values obtained from the different fits at a relatively lower magnitude of Vg, which can be ascribed to the background MR (as will be discussed further below). Nevertheless, both cases indicate that the ballistic contribution, defined as δ σ xx b = ne μ ' μ D with μDμ(T = 0K), has positive sign and therefore results in a partial cancelation of the diffusion correction. This is consistent with the prediction that the influence of e-e interactions is weakened in systems with long-range scattering potentials.

Figure 5
figure 5

ρ xx as a function of B2for V g = −0.125 (a), −0.145 (b), and−0.165 (c) V. The straight lines are provided as a guide to the eye to show the quadratic dependence on B.

Figure 6
figure 6

Renormalized mobility μ ′ as a function of T for V g = −0.125 (a), −0.145 (b), and−0.165 (c) V. The red and blue symbols denote the results obtained from the fits according to Equations 3 and 4, respectively. The insets are the zoom-ins of low-T results. The dotted lines represent the linear extrapolation of straight lines at T > 4 K.

At high magnetic fields B > 1/μD, semiclassical effects should affect the background resistance, resulting in either positive or negative MR [40, 41]. Therefore, it is not possible to obtain reliable values for μ′ from the first method. Here we use the value of μ′(T = 0K), obtained by linearly extrapolating the high-T results from the second method to T = 0 K [27, 34], to estimate μD and so as to allow a discussion on the role of the non-oscillatory background. As demonstrated in Figure 6, the estimated values of μD are 4.59, 3.79, and 2.89 m2/Vs for Vg = −0.125, −0.145, and −0.165 V, respectively, from which the corresponding ratios of μD/μq (5.22, 4.51, and 3.75) are determined with μq obtained by analyzing the amplitudes of SdH oscillations as shown in Figure 3. Since μq counts all scattering events whereas μD is sensitive only to large-angle ones, we can deduce the predominant scattering mechanism in a 2DES from the value of μD/μq[4244]. We can see from Figure 6 that both methods give the same results at low T for Vg = −0.165 V, implying that the influence of background MR is diminished as the amount of short-range scattering potential is increased. In what follows, we will focus on the issue about direct I-QH transitions.

Huckestein has suggested that the direct I-QH transition can be identified as a crossover from weak localization to the onset of Landau quantization, resulting in a strong reduction of the conductivity. The field B ~ 1/μ separates these two regions which are characterized by opposite T dependences and are characterized by ρxx ~ ρxy. In his argument, μ is taken to be the transport mobility. Nevertheless, recent experimental results [1113] demonstrate that different mobilities should be introduced to understand transport near a direct I-QH transition; the observed direct I-QH transition can be irrelevant to Landau quantization, while Landau quantization does not always cause the formation of QH states. Furthermore, it has already been demonstrated in various kinds of 2DES that the crossing point ρxx = ρxy can occur before or after the appearance of the T-independent point that corresponds to a direct I-QH transition. Moreover, the strongly T-dependent Hall slope induced by e-e interactions may affect the position of ρxx = ρxy at different T. As shown in Figure 2b for Vg = −0.145 V, the direct I-QH transition characterized by an approximately T-independent crossing point Bc in ρxx does occur at the field where ρxx ~ ρxy even though ρxy slightly depends on T. In addition, the inverse of the estimated Drude mobility 1/μD ~ 0.26 T is found to be close to Bc. To this extent, Huckestein's model seems to be reasonable. However, we can see that there are no apparent oscillations in ρxx around Bc and that the onset of strong localization occurs at B > 1.37 T, as characterized by a well-quantized ν = 2 Hall plateau and vanishing ρxx with increasing B, more than five times larger than Bc. In order to test the validity of the relation ρxx ~ ρxy at Bc, different gate voltages were applied to vary the effective amount of disorder and carrier density in the 2DES. As shown in Figure 2a, by increasing Vg to −0.125 V, ρxx becomes smaller than ρxy at Bc ~ 0.26 T, while ρxx ~ ρxy at a smaller field of approximately 0.21 T, which is shown to be close to 1/μD ~ 0.22 T rather than Bc. Moreover, by decreasing Vg to −0.165 V, ρxx ~ ρxy appears at B ~ 0.33 T which is larger than Bc ~ 0.29 T, as shown in Figure 2c. The inverse Drude mobility 1/μD ~ 0.35 is also found to be close to the field where ρxx ~ ρxy under this gate voltage. In all three cases, the crossings of σxx and σxy coincide with those of ρxx and ρxy, as shown in Figure 2 for each Vg. Therefore, our studies suggest that the field where ρxx ~ ρxy is governed by 1/μD and does not always correspond to that responsible for a direct I-QH transition as the influence of e-e interactions is not significant. As a result, ρxx ~ ρxy can occur on both sides of Bc as seen clearly in Figure 2d.

Interestingly, in the crossover from SdH oscillations to the QH state, we observe additional T-independent points, labeled by circles in Figure 2 for each Vg, other than the one corresponding to the onset of strong localization. As shown in Figure 2a for Vg = −0.125 V, the resistivity peaks at around B = 0.73 and 1.03 T appear to move with increasing T, a feature of the scaling behavior [7] of standard QH theory around the crossing points B = 0.70 and 0.96 T, respectively. Therefore, survival of the SdH theory for 0.46 T ≤ B ≤ 1.03 T reveals that semiclassical metallic transport may coexist with quantum localization. The superimposed background MR may be the reason for this coexistence, which is demonstrated by the upturned deviation from the parabolic dependence as shown in Figure 2a [45]. Therefore, it is reasonable to attribute the overestimated μ′ shown by the blue symbols in Figure 5a to the influence of the background MR. Similar behavior can also be found for Vg = −0.145 V even though spin splitting is unresolved, indicating that the contribution of background MR mostly comes from semiclassical effects. However, such a crossing point cannot be observed for Vg = −0.165 V since there is no clear separation between extended and localized states with strong disorder. Only a single T-independent point corresponding to the onset of strong localization occurs at B = 1.12 T.

In order to check the validity of our present results, further experiments were performed on a device (H597) with nominally T-independent Hall slope at different applied gate voltages [27]. As shown in Figure 7a for Vg = −0.05 V, weakly insulating behavior occurs as B < 0.62 T ≡ Bc, which corresponds to the direct I-QH transition since there is no evidence of the ν = 1 or ν = 2 QH state near Bc. The crossing of ρxx and ρxy is found to occur at B ~ 0.5 T which is smaller than Bc. As we decrease Vg to −0.1 V, thereby increasing the effective amount of disorder in the 2DES, the relative positions between these two fields remain the same as shown in Figure 7b. Nevertheless, it can be observed that ρxy tends to move closer to ρxx with decreasing Vg. This may be quantified by defining the ratio ρxy/ρxx at Bc, whose value is 1.57 and 1.31 for Vg = −0.05 and −0.1 V, respectively.

Figure 7
figure 7

ρ xx and ρ xy as functions of B at various T ranging from 0. 3 to 2 K. For (a) Vg = −0.05 V and (b) Vg = −0.1 V.

The interaction-induced parabolic NMR can be observed at both gate voltages. This result, together with the negligible T dependence of the Hall slope as shown in Figure 8a, implies that the ballistic part of the e-e interactions dominates as mentioned above. Therefore, by analyzing the observed parabolic NMR and corresponding Hall conductivity with Equations 4 and 3, respectively, we can obtain the renormalized transport mobilities μ′ at each measured T. Again, the estimated μ′ obtained by different methods as shown using different symbols in Figure 9 do not coincide with each other. It has already been demonstrated that the background MR can validate the SdH theory at B > 1/μq for Vg = −0.075 V in [27]. However, as shown in Figure 9c for Vg = −0.1 V, 1/μq ~ 1.67 T is found to be close to the crossing point in ρxx at B ~ 1.63 T, which corresponds to the ν = 4 to ν = 2 QH plateau-plateau transition. Therefore, it is reasonable to attribute the discrepancy of μ′ obtained by different methods to the background MR. However, we can see that the value of μ′ is underestimated by using the first method, which is different from that in sample LM4640 with the overestimated result. Our experimental results in conjunction with existing reports [37, 4548] suggest that a detailed treatment of the background MR is required. Moreover, the role of spin splitting does not seem to be significant in our system [4951].

Figure 8
figure 8

R H and ln( Δρ xx ( B , T )/ D ( B , T )). (a) RH as a function of T for both gate voltages. ln(Δρxx(B, T)/D(B, T)) as a function of 1/B is shown in (b) and (c) for Vg = −0.05 and −0.1 V, respectively. The dotted lines are the fits to Equation 1.

Figure 9
figure 9

μ′ as a function of T. For (a) Vg = 0 V, (b) Vg = −0.05 V, (c) Vg = −0.075 V, and (d) Vg = −0.1 V. The symbols are the same as those used in Figure 6.

The inverse Drude mobilities 1/μD estimated by the same procedures are 0.38, 0.46, 0.53, and 0.63 T for Vg = 0, −0.05, −0.075, and −0.1 V, respectively. We can see clearly that 1/μD deviates from the crossing of ρxx and ρxy (0.35, 0.43, 0.47, and 0.54 T for the corresponding Vg) as the applied gate voltage is decreased. The enhancement of background disorder with decreasing Vg may be the reason for such a discrepancy which can be deduced from the ratio μD/μq (4.27, 3.32, 2.92, and 2.65 for the corresponding Vg). The underlying physics is that the interference-induced e-e interactions are regained as a sufficient amount of short-range scattering potential is introduced, which leads to increased electron backscattering. Moreover, the parabolic NMR extending well below 1/μD, as shown in Figure 7, provides another evidence for the recovery of e-e interactions since in a 2DES dominated by a long-range scattering potential, it occurs only as B > 1/μD. We hope that our results will stimulate further investigations to fully understand the evolution of extended states near μDB = 1 in a disordered 2DES both experimentally and theoretically.


In conclusion, we have studied magnetotransport in gated two-dimensional electron systems. By varying the effective amount of disorder and the carrier density through different applied gate voltages, we observe that the crossing of ρxx and ρxy is governed by the inverse of the Drude mobility 1/μD and can occur for B > Bc, B < Bc, and B ~ Bc where Bc corresponds to the direct I-QH transition as the influence of e-e interactions is not significant. However, such a criterion breaks down when a sufficient amount of disorder is introduced, which leads to the recovery of interference-induced e-e interactions. Moreover, our results demonstrate that the magneto-oscillations following the semiclassical SdH theory can coexist with quantum localization as a result of the background MR, and the onset of strong localization occurs at a much higher field than either Bc or 1/μD. Therefore, in order to obtain a thorough understanding of the ground state of a weakly interacting 2DES, it is essential to eliminate the influence of e-e interactions as much as possible.


  1. Lee PA, Ramakrishnan TV: Disordered electronic systems. Rev Mod Phys 1985, 57: 287.

    Article  Google Scholar 

  2. Song SH, Shahar D, Tsui DC, Xie YH, Monroe D: New universality at the magnetic field driven insulator to integer quantum Hall effect transitions. Phys Rev Lett 1997, 78: 2200. 10.1103/PhysRevLett.78.2200

    Article  Google Scholar 

  3. Jiang HW, Johnson CE, Wang KL, Hannahs ST: Observation of magnetic-field-induced delocalization: transition from Anderson insulator to quantum Hall conductor. Phys Rev Lett 1993, 71: 1439. 10.1103/PhysRevLett.71.1439

    Article  Google Scholar 

  4. Hughes RJF, Nicholls JT, Frost JEF, Linfield EH, Pepper M, Ford CJB, Ritchie DA, Jones GAC, Kogan E, Kaveh M: Magnetic-field-induced insulator-quantum Hall-insulator transition in a disordered two-dimensional electron gas. J Phys Condens Matter 1994, 6: 4763. 10.1088/0953-8984/6/25/014

    Article  Google Scholar 

  5. Lee CH, Chang YH, Suen YW, Lin HH: Magnetic-field-induced insulator-quantum Hall conductor-insulator transitions in doped GaAs/AlxGa1-xAs quantum wells. Phys Rev B 1997, 56: 15238. 10.1103/PhysRevB.56.15238

    Article  Google Scholar 

  6. Smorchkova IP, Samarth N, Kikkawa JM, Awschalom DD: Giant magnetoresistance and quantum phase transitions in strongly localized magnetic two-dimensional electron gases. Phys Rev B 1998, 58: R4238. 10.1103/PhysRevB.58.R4238

    Article  Google Scholar 

  7. Huang CF, Chang YH, Lee CH, Chou HT, Yeh HD, Liang C-T, Chen YF, Lin HH, Cheng HH, Hwang GJ: Insulator-quantum Hall conductor transitions at low magnetic field. Phys Rev B 2002, 65: 045303.

    Article  Google Scholar 

  8. Kim G-H, Liang C-T, Huang CF, Lee MH, Nicholls JT, Ritchie DA: Insulator-quantum Hall transitions in two-dimensional electron gas containing self-assembled InAs dots. Physica E 2003, 17: 292.

    Article  Google Scholar 

  9. Kim G-H, Liang C-T, Huang CF, Nicholls JT, Ritchie DA, Kim PS, Oh CH, Juang JR, Chang YH: From localization to Landau quantization in a two-dimensional GaAs electron system containing self-assembled InAs quantum dots. Phys Rev B 2004, 69: 073311.

    Article  Google Scholar 

  10. Huang T-Y, Juang JR, Huang CF, Kim G-H, Huang C-P, Liang C-T, Chang YH, Chen YF, Lee Y, Ritchie DA: On the low-field insulator-quantum Hall conductor transitions. Physica E 2004, 22: 240. 10.1016/j.physe.2003.11.258

    Article  Google Scholar 

  11. Huang TY, Liang C-T, Kim G-H, Huang CF, Huang CP, Lin JY, Goan HS, Ritchie DA: From insulator to quantum Hall liquid at low magnetic fields. Phys Rev B 2008, 78: 113305.

    Article  Google Scholar 

  12. Chen KY, Chang YH, Liang C-T, Aoki N, Ochiai Y, Huang CF, Lin L-H, Cheng KA, Cheng HH, Lin HH, Wu JY, Lin S-D: Probing Landau quantization with the presence of insulator–quantum Hall transition in a GaAs two-dimensional electron system. J Phys Condens Matter 2008, 20: 295223. 10.1088/0953-8984/20/29/295223

    Article  Google Scholar 

  13. Lo ST, Chen KY, Lin TL, Lin LH, Luo DS, Ochiai Y, Aoki N, Wang Y-T, Peng ZF, Lin Y, Chen JC, Lin SD, Huang CF, Liang CT: Probing the onset of strong localization and electron–electron interactions with the presence of a direct insulator–quantum Hall transition. Solid State Commun 1902, 2010: 150.

    Google Scholar 

  14. Liu DZ, Xie XC, Niu Q: Weak field phase diagram for an integer quantum Hall liquid. Phys Rev Lett 1996, 76: 975. 10.1103/PhysRevLett.76.975

    Article  Google Scholar 

  15. Sheng DN, Weng ZY: Phase diagram of the integer quantum Hall effect. Phys Rev B 2000, 62: 15363. 10.1103/PhysRevB.62.15363

    Article  Google Scholar 

  16. Huckestein B: Quantum Hall effect at low magnetic fields. Phys Rev Lett 2000, 84: 3141. 10.1103/PhysRevLett.84.3141

    Article  Google Scholar 

  17. Hanein Y, Nenadovic N, Shahar D, Shtrikman H, Yoon I, Li CC, Tsui DC: Linking insulator-to-metal transitions at zero and finite magnetic fields. Nature 1999, 400: 735. 10.1038/23419

    Article  Google Scholar 

  18. Clarke WR, Yasin CE, Hamilton AR, Micolich AP, Simmons MY, Muraki K, Hirayama Y, Pepper M, Ritchie DA: Impact of long- and short-range disorder on the metallic behaviour of two-dimensional systems. Nat Phys 2008, 4: 55. 10.1038/nphys757

    Article  Google Scholar 

  19. Ilani S, Martin J, Teitelbaum E, Smet JH, Mahalu D, Umansky V, Yacoby A: The microscopic nature of localization in the quantum Hall effect. Nature 2004, 427: 328. 10.1038/nature02230

    Article  Google Scholar 

  20. Amado M, Diez E, Lopez-Romero D, Rossella F, Caridad JM, Dionigi F, Bellani V, Maude DK: Plateau–insulator transition in graphene. New J Phys 2010, 12: 053004. 10.1088/1367-2630/12/5/053004

    Article  Google Scholar 

  21. Fowler AB, Fang FF, Howard WE, Stiles PJ: Magneto-oscillatory conductance in silicon surfaces. Phys Rev Lett 1966, 16: 901. 10.1103/PhysRevLett.16.901

    Article  Google Scholar 

  22. Ando T: Theory of quantum transport in a two-dimensional electron system under magnetic fields. IV. Oscillatory conductivity. J Phys Soc Jpn 1974, 37: 1233. 10.1143/JPSJ.37.1233

    Article  Google Scholar 

  23. Coleridge PT, Zawadzki P, Sachrajda AS: Peak values of resistivity in high-mobility quantum-Hall-effect samples. Phys Rev B 1994, 49: 10798. 10.1103/PhysRevB.49.10798

    Article  Google Scholar 

  24. Martin GW, Maslov DL, Reizer MY: Quantum magneto-oscillations in a two-dimensional Fermi liquid. Phys Rev B 2003, 68: 241309.

    Article  Google Scholar 

  25. Hang DR, Huang CF, Cheng KA: Probing semiclassical magneto-oscillations in the low-field quantum Hall effect. Phys Rev B 2009, 80: 085312.

    Article  Google Scholar 

  26. Huang TY, Liang C-T, Kim G-H, Huang CF, Huang CP, Ritchie DA: Probing two-dimensional metallic-like and localization effects at low magnetic fields. Physica E 2010, 42: 1142. 10.1016/j.physe.2009.11.049

    Article  Google Scholar 

  27. Lo S-T, Wang Y-T, Bohra G, Comfort GE, Lin T-Y, Kang M-G, Strasser G, Bird JP, Huang CF, Lin L-H, Chen JC, Liang C-T: Insulator, semiclassical oscillations and quantum Hall liquids at low magnetic fields. J Phys Condens Matter 2012, 24: 405601. 10.1088/0953-8984/24/40/405601

    Article  Google Scholar 

  28. Gao KH, Yu G, Zhou YM, Wei LM, Lin T, Shang LY, Sun L, Yang R, Zhou WZ, Dai N, Chu JH, Austing DG, Gu Y, Zhang YG: Insulator-quantum Hall conductor transition in high electron density gated InGaAs/InAlAs quantum wells. J Appl Phys 2010, 108: 063701. 10.1063/1.3486081

    Article  Google Scholar 

  29. Dubi Y, Meir Y, Avishai Y: Quantum Hall criticality, superconductor-insulator transition, and quantum percolation. Phys Rev B 2005, 71: 125311.

    Article  Google Scholar 

  30. Dubi Y, Meir Y, Avishai Y: Unifying model for several classes of two-dimensional phase transition. Phys Rev Lett 2005, 94: 156406.

    Article  Google Scholar 

  31. Zala G, Narozhny BN, Aleiner IL: Interaction corrections at intermediate temperatures: longitudinal conductivity and kinetic equation. Phys Rev B 2001, 64: 214204.

    Article  Google Scholar 

  32. Gornyi IV, Mirlin AD: Interaction-induced magnetoresistance: from the diffusive to the ballistic regime. Phys Rev Lett 2003, 90: 076801.

    Article  Google Scholar 

  33. Gornyi IV, Mirlin AD: Interaction-induced magnetoresistance in a two-dimensional electron gas. Phys Rev B 2004, 69: 045313.

    Article  Google Scholar 

  34. Minkov GM, Germanenko AV, Rut OE, Sherstobitov AA, Larionova VA, Bakarov AK, Zvonkov BN: Diffusion and ballistic contributions of the interaction correction to the conductivity of a two-dimensional electron gas. Phys Rev B 2006, 74: 045314.

    Article  Google Scholar 

  35. Minkov GM, Germanenko AV, Rut OE, Sherstobitov AA, Zvonkov BN: Renormalization of hole-hole interaction at decreasing Drude conductivity: gated GaAs/InxGa1−xAs/GaAs heterostructures. Phys Rev B 2007, 76: 165314.

    Article  Google Scholar 

  36. Paalanen MA, Tsui DC, Hwang JCM: Parabolic magnetoresistance from the interaction effect in a two-dimensional electron gas. Phys Rev Lett 1983, 51: 2226. 10.1103/PhysRevLett.51.2226

    Article  Google Scholar 

  37. Li L, Proskuryakov YY, Savchenko AK, Linfield EH, Ritchie DA: Magnetoresistance of a 2D electron gas caused by electron interactions in the transition from the diffusive to the ballistic regime. Phys Rev Lett 2003, 90: 076802.

    Article  Google Scholar 

  38. Simmons MY, Hamilton AR, Pepper M, Linfield EH, Rose PD, Ritchie DA: Weak localization, hole-hole interactions, and the “metal”-insulator transition in two dimensions. Phys Rev Lett 2000, 84: 2489. 10.1103/PhysRevLett.84.2489

    Article  Google Scholar 

  39. Hilke M, Shahar D, Song SH, Tsui DC, Xie YH: Phase diagram of the integer quantum Hall effect in p-type germanium. Phys Rev B 2000, 62: 6940. 10.1103/PhysRevB.62.6940

    Article  Google Scholar 

  40. Mirlin AD, Polyakov DG, Evers F, Wölfle P: Quasiclassical negative magnetoresistance of a 2D electron gas: interplay of strong scatterers and smooth disorder. Phys Rev Lett 2001, 87: 126805.

    Article  Google Scholar 

  41. Renard V, Kvon ZD, Gusev GM, Portal JC: Large positive magnetoresistance in a high-mobility two-dimensional electron gas: interplay of short- and long-range disorder. Phys Rev B 2004, 70: 033303.

    Article  Google Scholar 

  42. Harrang JP, Higgins RJ, Goodall RK, Jay PR, Laviron M, Delescluse P: Quantum and classical mobility determination of the dominant scattering mechanism in the two-dimensional electron gas of an AlGaAs/GaAs heterojunction. Phys Rev B 1985, 32: 8126. 10.1103/PhysRevB.32.8126

    Article  Google Scholar 

  43. Das Sarma S, Stern F: Single-particle relaxation time versus scattering time in an impure electron gas. Phys Rev B 1985, 32: 8442. 10.1103/PhysRevB.32.8442

    Article  Google Scholar 

  44. Coleridge PT, Stoner R, Fletcher R: Low-field transport coefficients in GaAs/Ga1-xAlxAs heterostructures. Phys Rev B 1989, 39: 1120. 10.1103/PhysRevB.39.1120

    Article  Google Scholar 

  45. Gao KH, Zhou WZ, Zhou YM, Yu G, Lin T, Guo SL, Chu JH, Dai N, Gu Y, Zhang YG, Austing DG: Magnetoresistance in high-density two-dimensional electron gas confined in InAlAs/InGaAs quantum well. Appl Phys Lett 2009, 94: 152107. 10.1063/1.3119664

    Article  Google Scholar 

  46. Hang DR, Liang C-T, Juang JR, Huang T-Y, Hung WK, Chen YF, Kim G-H, Lee J-H, Lee J-H: Electrically detected and microwave-modulated Shubnikov-de Haas oscillations in an Al0.4Ga0.6N/GaN heterostructure. J Appl Phys 2003, 93: 2055. 10.1063/1.1539286

    Article  Google Scholar 

  47. Juang JR, Huang T-Y, Chen T-M, Lin M-G, Lee Y, Liang C-T, Hang DR, Chen YF, Chyi J-I: Transport in a gated Al0.18Ga0.82N/GaN electron system. J Appl Phys 2003, 94: 3181. 10.1063/1.1594818

    Article  Google Scholar 

  48. Chen JH, Lin JY, Tsai JK, Park H, Kim G-H, Youn D, Cho HI, Lee EJ, Lee JH, Liang C-T, Chen YF: Experimental evidence for Drude-Boltzmann-like transport in a two-dimensional electron gas in an AlGaN/GaN heterostructure. J Korean Phys Soc 2006, 48: 1539.

    Google Scholar 

  49. Cho KS, Huang T-Y, Huang CP, Chiu YH, Liang C-T, Chen YF, Lo I: Exchange-enhanced g-factors in an Al0.25Ga0.75N/GaN two-dimensional electron system. J Appl Phys 2004, 96: 7370. 10.1063/1.1815390

    Article  Google Scholar 

  50. Cho KS, Liang C-T, Chen YF, Tang YQ, Shen B: Spin-dependent photocurrent induced by Rashba-type spin splitting in Al0.25Ga0.75N/GaN heterostructures. Phys Rev B 2007, 75: 085327.

    Article  Google Scholar 

  51. Lin S-K, Wu KT, Huang CP, Liang C-T, Chang YH, Chen YF, Chang PH, Chen NC, Chang C-A, Peng HC, Shih CF, Liu KS, Lin TY: Electron transport in In-rich InxGa1-xN films. J Appl Phys 2005, 97: 046101. 10.1063/1.1847694

    Article  Google Scholar 

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This work was funded by National Taiwan University (grant no. 102R7552-2).

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Correspondence to Chi-Te Liang.

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The authors declare that they have no competing interests.

Authors’ contributions

STL and YTW performed the experiments. GS and SDL prepared the devices. YFC and CTL coordinated the project. STL, JPB, and CTL drafted the paper. All the authors read and approved the final version of the manuscript.

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Lo, ST., Wang, YT., Lin, SD. et al. Tunable insulator-quantum Hall transition in a weakly interacting two-dimensional electron system. Nanoscale Res Lett 8, 307 (2013).

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  • Hall effect
  • Magnetoresistance
  • Electrons
  • Direct insulator-quantum hall transition