Abstract
A direct insulatorquantum Hall (IQH) transition corresponds to a crossover/transition from the insulating regime to a high Landau level filling factor ν > 2 QH state. Such a transition has been attracting a great deal of both experimental and theoretical interests. In this study, we present three different twodimensional electron systems (2DESs) which are in the vicinity of nanoscaled scatterers. All these three devices exhibit a direct IQH transition, and the transport properties under different nanaoscaled scatterers are discussed.
Introduction
The simultaneous presence of disorder and a strong enough magnetic field B can lead to a wide variety of interesting physical phenomena. For example, the integer quantum Hall effect is one of the most exciting effects in twodimensional electron systems (2DES), in which the electrons are usually confined in layers of the nanoscale [1]. In an integer quantum Hall (QH) state, the current is carried by the onedimensional edge channels because of the localization effects. It has been shown that with sufficient amount of disorder, a 2DES can undergo a Binduced insulator to quantum Hall transition [25]. Experimental evidence for such an insulatorquantum Hall (IQH) transition is an approximately temperature (T)independent point in the measured longitudinal resistivity of a 2DES [35]. The IQH transition continues to attract a great deal of interest both experimentally and theoretically as it may shed light on the fate of extended states [610], the true ground state of a noninteracting 2DES [2], and a possible metalinsulator transition in 2D [11,12].
It is worth pointing out that in order to observe an IQH transition separating the zerofield insulator from the QH liquid, one needs to deliberately introduce strong disorder within a 2DES. The reason for this is that the localization length needs to be shorter than the sample size. In the study by Jiang and coworkers [2], a 2DES without a spacer layer in which strong Coulomb scattering exists was used. Wang et al. utilized a 30nmthick heavily doped GaAs layer so as to allow the positively charged Si atoms to introduce longrange random potential in the 2DES [3]. Hughes et al. have shown that when a Sidoped plane was incorporated into a 550nmthick GaAs film, a deep potential well can form in which the 2DES is confined close to the ionized donors and is therefore highly disordered [4]. It has been shown that by deliberately introducing nanoscaled InAs quantum dots [13] in the vicinity of a modulationdoped GaAs/AlGaAs heterostructure, a strongly disordered 2DES which shows an IQH transition can be experimentally realized [14,15].
The transition/crossover from an insulator to a QH state of the filling factor ν > 2 in an ideal spinless 2DES can be denoted as the direct IQH transition [1619]. Such a transition has been attracting a great deal of interest and remains an unsettled issue. Experimental [1619] and theoretical results [9,10] suggest that such a direct transition can occur, and it is a quantum phase transition. However, Huckestein [20] has argued that such a direct transition is not a quantum phase transition, but a narrow crossover in B due to weak localization to Landau quantization.
In this study, the authors compare three different electron systems containing nanoscaled scatterers which all show a direct IQH transition. The first sample is a GaAs 2DES containing selfassembled nanoscaled InAs quantum dots [13,14,2123].
The second one is a 2DES in a nominally undoped AlGaN/GaN heterostructure [2433] grown on Si substrate [33,34]. Such a GaNbased electron system can be affected by nanoscaled dislocation and impurities [35]. Finally, experimental results on the third sample, a deltadoped GaAs/AlGaAs quantum well with additional modulation doping [36,37], will be presented. All the experimental results on the three completely different samples show that the direct IQH transition does not occur with the onset of strong localization due to Landau quantization [20,38]. Therefore, in order to obtain a thorough understanding of the direct IQH transition, further studies are required.
Experimental details
Figure 1a,b,c show the structures of the three devices, Sample A, Sample B, and Sample C, considered in this study. Sample A is a GaAs/AlGaAs 2DES containing selfassembled InAs quantum dots. Sample B is an AlGaN/GaN heterostructure grown on Si. Such a system is fully compatible with Si CMOS technology and is thus of great potential applications. Sample C is a deltadoped quantum well with additional deltadoping. Since the electrons in the quantum well in sample B are in close proximity of nanoscaled dislocation and impurities, the 2DES is strongly influenced by these nanoscaled scatterers. In fact, these scatterers provide scattering which is required for observing the IQH transition [16]. On the other hand, the scatterings in samples A and C are mainly due to the selfassembled quantum dots and the deltadoping in the quantum well, respectively. Recent studies focussing on alloy disorder in Al_{x}Ga_{1x}As/GaAs heterostructure [3941] have shown that 2DESs influenced by shortrange disorder provides an excellent opportunity to connect the Anderson localization theory with real experimental systems [41]. Moreover, the nature of disorder may affect scaling behavior in the plateauplateau (PP) transition at high B [3941], and the PP and IQH transitions may be considered as the same universality class [42]. Therefore, it may be interesting to investigate the direct IQH transitions under different scattering types at low magnetic fields. In this article, such lowfield direct transitions in samples A, B, and C are compared.
Figure 1. Schematic diagrams showing the structure of (a) Sample A, (b) Sample B, and (c) Sample C.
Figure 2 shows a TEM image of the wafer for fabricating Sample A. Very uniform nanoscaled InAs quantum dots can be seen. These nanoscattering centers provide strong scattering in the vicinity of the 2DES in the GaAs. The dimensions of the quantum dot are estimated to be 20 nm in diameter and 4 nm in height. Experiments were performed in a toploading He3 cryostat equipped with a superconductor magnet. Fourterminal resistance measurements were performed using standard phasesensitive lockin techniques.
Figure 2. A planeview of TEM image of the wafer which was cut to fabricate sample A.
Results and discussions
Figure 3 shows the longitudinal magnetoresistivity measurements on Sample A as a function of B at various temperatures. It can be seen that at a crossing field Bc = 0.9 T, ρ_{xx }is approximately Tindependent. For B <B_{c}, ρ_{xx}decreases with increasing temperature, characteristics of an insulating regime [16]. For B > B_{c}, ρ_{xx}increases with increasing temperature, and therefore the 2DES is in the quantum Hall regime. As the 2DES enters the ν = 4 QH state from the insulating regime, a direct 04 transition where the symbol 0 corresponds to the insulator has been observed. It is worth pointing out that before the 2DES enters the ν = 4 QH state, resistance oscillations due to Landau quantization in the insulating regime have already been observed [15,19,21]. Therefore, the experimental results of this study clearly demonstrate that the crossover from localization from Landau quantization actually covers a wide range of magnetic field, in sharp contrast to Huckestein's argument [1921].
Figure 3. ρ_{xx}(B) at various temperatures ranging from 0.25 to 2.85 K (Sample A).
As mentioned earlier, a GaNbased electron system can be affected by nanoscaled dislocation and impurities. It is therefore interesting to study such a system. Figure 4 shows magnetoresistance measurements on Sample B as a function of magnetic field at different temperatures. The data deviate slightly from the expected symmetric behavior, i.e., R(B) = R(B). The reason for this could be due to slight misalignment of the voltage probes. Nevertheless, it can be seen that at B_{c} = 11 Tand B_{c} = 11 T, the measured resistances are approximately temperature independent. The corresponding Landau level filling factor is about 50 in this case. Therefore, a direct 050 transition has been observed. Note that even at the highest attainable field of approximately 15 T, there is no sign of resistance oscillations due to the moderate mobility of our GaN system. Therefore, the experimental results of this study clearly demonstrate that the observed direct IQH transition is irrelevant to Landau quantization. Therefore, the onset of Landau quantization does not necessarily accompany the direct IQH transition, inconsistent with Huckestein's model [20].
Figure 4. ρ_{xx}(B) at various temperatures ranging from 0.28 to 20 K (Sample B).
Figure 5 shows magnetoresistance measurements on Sample C as a function of magnetic field at various temperatures. It can be seen that the 2DES undergoes a 08 transition characterized by an approximately temperatureindependent point in ρ _{xx}at the crossing field B_{c}. Near the crossing field, ρ_{xx}is very close to ρ_{xy}though ρ_{xy}shows a weak T dependence. For B < B_{c}, no resistance oscillation is observed. At first glance, our experimental results are consistent with Huckestein's model. However, it is noted that Landau quantization should be linked with quantum mobility, not classical Drude mobility [36]. Moreover, the observed oscillations for B > B_{c} do not always correspond to formation of quantum Hall states. As mentioned in our previous study [36], the observed oscillations can be well approximated by conventional Shubnikovde Haas (SdH) formalism. It is noted that the SdH formula is derived without considering quantum localization effects which give rise to formation of quantum Hall state. Therefore, quantum localization effects are not significant in the system under this study. Actually, as shown in Figure 6, the crossing point in σ_{xy} at around 7.9 Tmay correspond to the extended states due to the onset of the strong localization effects. Therefore, in this study, the onset of strong localization actually occurs at a magnetic field approximately 4 Thigher than the crossing point.
Figure 5. ρ_{xx}(B) at various temperatures ranging from 0.3 to 4 K (Sample C). ρxx at T = 0.3 K and T = 4 K are shown.
Figure 6. Converted σ_{xx}(B) and σ_{xy}(B) at various temperatures ranging from 0.3 to 4 K (Sample C).
It has been suggested that by converting the measured resistivities into longitudinal and Hall conductivities, it is possible to shed more light on the observed IQH transition [5]. Figure 6 shows such results at various temperatures. Interestingly, for B < 5 T, _{σ}_{xy}is nominally T independent. Such data are consistent with electronelectron interaction effects. Over the whole measurement range, σ_{xx}decreases with increasing T, consistent with electronelectron interaction effects. Unlike σ_{xy}, σ_{xx}shows a significant Tdependence.
By inspecting the conductivies, previously the authors have studied the renormalized mobility [43] of a GaNbased 2DES at high temperatures (Sample B) [44]. It is therefore interesting to study such a mobility for both Sample A and Sample C. It has been suggested the electronelectron interaction effects can renormalize the mobility μ' given by
Figure 7 and the inset to Figure 7 show σ_{xy} and σ_{xx} , together with fits to Equations 1 and 2 over limited ranges for Sample C, respectively. From the fits, it is possible to determine the respective renormalized mobilites as a function of temperature as shown in Figure 8a for Sample C and in Figure 8b for Sample A. The renormalized mobility calculated using Equation 1 is only slightly larger than that using Equation 2. It may be possible that different mobilities should be taken into account to understand the direct IQH transition [37,43,45].
Figure 7. σ_{xy}(B) and the fit to Equation 1 for 0 <B < 3.5 T. The inset shows σ_{xx}(B) and the fit to Equation (2) for 1 T <B < 3.5 T.
Figure 8. Calculated renormalized mobilities due to electronelectron interaction effects using Equations (1) and (2) for (a) Sample C and (b) Sample A, respectively.
Conclusions
In conclusion, the authors have presented studies on three completely different electron systems. In these three samples, the nanoscaled scatterers, in close proximity of the 2DES, provide necessary disorder for observing the direct IQH transition. In these studies, it has been shown that the crossover from localization to Landau quantization actually covers a wide range of magnetic field. Moreover, the observed direct IQH transition is not necessarily linked with Landau quantization as no resistance oscillations are observed even up to a magnetic field 4 T higher than the crossing field. Most importantly, the onset of strong localization which gives rise to the formation of quantum Hall state does not correspond to the direct IQH transition. All these three pieces of experimental evidence show that a 2DES in the vicinity of nanoscaled scatterers is an ideal playground for studying the direct IQH transition. Furthermore, in order to obtain a thorough understanding of the underlying physics of the direct IQH transition, modifications of Huckestein's model [20] must be made.
Abbreviations
IQH: insulatorquantum Hall; SdH: Shubnikovde Haas; 2DESs: twodimensional electron systems.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
CTL, GHK and YHC coordinated the measurements on Sample A. CTL coordinated the measurements on Sample B. KYC performed the measurements on Sample B. JCC and YL coordinated the measurements on Sample C undertaken in Taiwan. YO and NA coordinated early measurements on Sample C in Japan. CTL, STL and CFH drafted the manuscript. LHL, YTW and DLS performed measurements on Sample C. SDL and DAR grew the MBE wafers. All authors read and approved the final manuscript.
Acknowledgements
This research was supported by the WCU (World Class University) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. R322008000102040). C.T.L. acknowledges financial support from the NSC (Grant no: NSC 992119M002018MY3). The authors would like to thank YiChun Su and JauYang Wu for providing help in the experiments.
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