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Tunable insulator-quantum Hall transition in a weakly interacting two-dimensional electron system
© Lo et al.; licensee Springer. 2013
Received: 7 May 2013
Accepted: 26 June 2013
Published: 3 July 2013
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.
where μq represents the quantum mobility, D(B, T) = 2π2kBm * T/ℏeB sinh (2π2kBm * T/ℏeB), 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 . In addition, as reported in , 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  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 with reduced Planck's constant (ℏ), 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 . 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 (kBTτ/ℏ < < 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 (kBTτ/ℏ > > 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.
The Hall slope RH (ρxy/B with Hall resistivity ρxy) now becomes T-dependent which is ascribed to the diffusion correction . 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.
While Equation 5 holds true in some experiments , 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 . 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 , 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
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[42–44]. 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 [11–13] 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  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 . 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.
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.
This work was funded by National Taiwan University (grant no. 102R7552-2).
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