N A N O E X P R E S S
Open Access
Tunable insulator-quantum Hall transition in a
weakly interacting two-dimensional electron
system
Shun-Tsung Lo
1, Yi-Ting Wang
2, Sheng-Di Lin
3, Gottfried Strasser
4, Jonathan P Bird
5, Yang-Fang Chen
1,2and Chi-Te Liang
1,2*Abstract
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 (ρxxandρxy) can be observed. Interestingly, by applying different gate
voltages, we demonstrate that such a crossing atρxx~ρxycan occur at a magnetic field higher, lower, or equal to
the temperature-independent point inρxxwhich corresponds to the direct insulator-quantum Hall transition. We
explicitly show thatρxx~ρxyoccurs at the inverse of the classical Drude mobility 1/μDrather 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.
Keywords: Hall effect; Magnetoresistance; Electrons; Direct insulator-quantum hall transition Background
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, lead-ing 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
in-creasing T [2]. It is useful to probe the intriguing fea-tures of this 2D insulating state by applying a magnetic field (B) perpendicular to the plane of a 2DES [2-4]. In particular, the direct transition from an insulator (I) to a high filling factor (v ≥ 3) quantum Hall (QH) state con-tinues to attract a great deal of both experimental [5-13] and theoretical [14-16] 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 back-ground 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. Previ-ously, we have studied a 2DES containing self-assembled InAs quantum dots [11], providing a predominantly short-range character to the disorder. We observed mul-tiple T-independent points in ρxx(B), indicating a series
of transitions between a low-field insulator and a QH state. The oscillatory amplitude of ρxx(B) was well fitted
by the Shubnikov-de Haas (SdH) theory [21-23], with amplitude given by
ΔρxxðB; TÞ ¼ Cexp −π=μqB
D B; Tð Þ; ð1Þ
where μq represents the quantum mobility, D(B, T) =
2π2kBm * T/ℏeB sinh (2π2kBm * T/ℏeB), and C is a
con-stant relevant to the value ofρxxat B = 0 T. The
obser-vation of the SdH oscillations suggests the possible existence of a Fermi-liquid metal. It should be pointed * Correspondence:[email protected]
1Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan
2Department of Physics, National Taiwan University, Taipei 106, Taiwan Full list of author information is available at the end of the article
© 2013 Lo et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
out that the SdH theory is derived by considering Lan-dau 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 demon-strated [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 sig-nificance of percolation has been revealed both experi-mentally [28] and theoretically [29,30]. Therefore, to fully understand the direct I-QH transition, further stud-ies 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 con-ductivity, 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 by4eDτℏ 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 interac-tions 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 phys-ics 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
modi-fying the Hall conductivity (σxy). On the other hand, in
the ballistic regime (kBTτ/ℏ > > 1), e-e interactions
con-tribute both to σxx and σxy, and effectively reduce to a
renormalization of the transport mobility. However, the situation is different for long-range potential fluctua-tions, which are usually dominant in high-quality GaAs-based heterostructures in which the dopants are separated from the 2D electron gas by an undoped spa-cer. 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/μDwith enhanced
return probability of scattered electrons, whereμD
repre-sents the Drude mobility [32,33]. Therefore, it is of great interest to study the direct insulator-quantum Hall tran-sition 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 kBTτ/ℏ ≈ 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 mobilityμ′ induced by the ballistic contribution and the diffusion correctionδσd xx,σxxis expressed as σxx¼ neμ 0 1þ μ02B2þ δσ d xx; ð2Þ σxy¼ neμ 02B 1þ μ02B2: ð3Þ
It directly follows that the ballistic contributionδσb xxis given byδσb
xx¼ ne μð 0−μDÞ; where n is the electron dens-ity andμDis the transport mobility derived in the Drude
model. After performing matrix inversion with the com-ponents given in Equations 2 and 3, the magnetoresist-anceρxx(B) takes the parabolic form [36,37]
ρxx≈ 1 neμ0− 1 neμ0 ð Þ2 1−μ02B 2δσd xx ð4Þ
The Hall slope RH (ρxy/B with Hall resistivity ρxy) now
becomes T-dependent which is ascribed to the diffusion correctionδσd
xx[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 inρxx,
ρxx∼ρxy ð5Þ
While Equation 5 holds true in some experiments [2], in others it has been found thatρxy can be significantly
higher than ρxxnear the direct I-QH transition [10,28].
On the other hand, ρxy can also be lower than ρxxnear
the direct I-QH transition in some systems [39]. There-fore, 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 disor-dered 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 interac-tions. We are able to tune the direct I-QH transition so
that the corresponding field for which Equation 5 is sat-isfied can be higher or lower than, or even equal, to the crossing field that corresponds to the direct I-QH transi-tion. Interestingly, we show that the inverse Drude mo-bility 1/μDis approximately equal to the field where ρxx
crossesρxy, 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.
Methods
A gated modulation-doped AlGaAs/GaAs heterostructure (LM4640) is used in our study. The following layer se-quence 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
centi-meter, 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 evap-oration. 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
tech-niques 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-developedν = 2 QH state manifests itself in the quantized ν = 2 Hall plateau and the associated vanishing ofρxx. In order to study the transition from an
insulator to a QH state, detailed results ofρxxandρxyat
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 ≡ Bcfor Vg=
−0.125, −0.145, and −0.165 V, respectively, since there is no signature ofν = 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 Bcwith respect to that corresponding to the crossing of
ρxxandρxyis not necessarily equal. Following the
transi-tion, magneto-oscillations superimposed on the back-ground 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
agree-ment 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
0.0 0.5 1.0 1.5 2.0 2.5 0 500 1000 1500 2000 2500 3000 0 4 8 12 16 1000 1500 2000 2500 V g = -0.125 V Vg = -0.145 V xx ( B = 0 T ) T (K) Vg = -0.165 V B (T) xx
()
0 2 4 6 8 10 12 14 16 xy(k
)
V g = -0.145 V 4840 4860 4880 4900 RH ( /T) RH ( /T) 5580 5600 5620 5640 V g = -0.165 V Vg = -0.145 V Vg = -0.125 V 1 10 6560 6600 6640 6680 RH ( /T) T (K)a
b
Figure 1 Temperature dependence. (a) Longitudinal and Hall resistivities (ρxxandρxy) 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 RHas a function of T at each Vgon a semi-logarithmic scale.
follows that we can obtain the quantum mobility μq
from the fits, which is expected to be an essential quan-tity regarding Landau quantization. The estimatedμqare
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-in-dependent 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.
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 measuredρxxto Equation 4, as shown by the
blue symbols in Figure 5, from which we can obtain both δσb
xx and δσdxx. The value of δσdxx 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 obtainedμ′, with an approximately linear de-pendence on T that is characteristic of the ballistic con-tribution 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
0.0 0.5 1.0 1.5 2.0 0 500 1000 1500 2000 2500 0.2 0.4 0.6 2000 2200 2400 xx ( ) B (T) 0.322 K 0.949 K 1.533 K B = 0.33 T xx
&
xy()
B (T)
0.322 K 0.535 K 0.739 K 0.949 K 1.125 K 1.321 K 1.533 K Vg = -0.165 V 0.0 0.5 1.0 1.5 2.0 0 900 1800 2700 0.1 0.2 0.3 0.4 1400 1470 1540 xx & xy () B (T) 0.326 K 0.928 K 1.566 K Vg = -0.145 V xx&
xy()
B (T)
0.326 K 0.533 K 0.736 K 0.928 K 1.126 K 1.333 K 1.566 K B = 0.26 T 0.0 0.4 0.8 1.2 1.6 2.0 0 400 800 1200 1600 2000 0.15 0.20 0.25 0.30 0.35 990 1020 1050 xx & xy () B (T) 0.325 K 0.934 K 1.552 K xx&
xy()
B (T) 0.325 K 0.554 K 0.731 K 0.934 K 1.129 K 1.337 K 1.552 K V g = -0.125 V B = 0.21 T 0.0 0.1 0.2 0.3 0.4 0.5 0.6 600 1200 1800 2400 Vg = -0.125 V Vg = -0.145 V Vg = -0.165 V xx&
xy()
B (T)
a
b
c
d
Figure 2 Detailed results ofρxxandρxyat lowT. The B dependences of ρxxandρxyat 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 Vgare 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 ofρxxandρxyare denoted by arrows. Other T-independent points are indicated by circles.
obtain μ′ for T > 4 K since there is no apparent para-bolic 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 para-bolic 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 Bmaxdenotes the
0.0 0.5 1.0 1.5 2.0 0.0 3.0x10-4 6.0x10-4 9.0x10-4 0.0 0.2 0.4 0.0 2.0x10-4 4.0x10-4 T = 0.326 K xy (S) B (T) T = 16 K xx
&
xy(S)
B (T) B = 0.26 T V g = -0.145 V 0.0 0.5 1.0 1.5 2.0 0.0 1.0x10-4 2.0x10-4 3.0x10-4 4.0x10-4 5.0x10-4 0.0 0.2 0.4 0.0 1.0x10-4 2.0x10-4 3.0x10-4 T = 0.322 K xy ( ) B (T) T = 16 K Vg = -0.165 V xx&
xy(S)
B (T) B = 0.33 T 0.0 0.4 0.8 1.2 1.6 2.0 0.0 3.0x10-4 6.0x10-4 9.0x10-4 1.2x10-3 0.0 0.2 0.4 0.0 2.0x10-4 4.0x10-4 6.0x10-4 T = 0.325 K xy (S) B (T) T = 16 K xx&
xy(S)
B (T) B = 0.21 T V g = -0.125 Va
b
c
Figure 3 Convertedσxx(B) and σxy(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 showσxy(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 ofσxxandσxy.
1.0 1.5 2.0 0 2 4 6 Vg = -0.125 V n = 1.25 x 1015 m-2 q = 0.88 m 2 /Vs 0.325 K 0.554 K 0.731 K 0.934 K 1.129 K 1.337 K 1.552 K
1/ B (T
-1)
ln(
xx/(
/sinh
)
1.0 1.5 2.0 2 4 6 n = 1.06 x 1015 m-2 q = 0.84 m 2 /Vs 0.326 K 0.533 K 0.736 K 0.928 K 1.126 K 1.333 K 1.566 K1/ B (1/T)
ln(
xx/(
/sinh
)
V g = -0.145 V 1.0 1.5 2.0 2 3 4 5 6 7 V g = -0.165 V q = 0.77 m 2 /Vs n = 9.01 x 1014 m-2 0.322 K 0.535 K 0.739 K 0.949 K 1.125 K 1.321 K 1.533 K1/ B (T
-1)
ln(
xx/(
/sinh
)
a
b
c
Figure 4 ln (Δρxx(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.
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 indi-cate that the ballistic contribution, defined as δσb
xx¼ ne 0.0 0.1 0.2 0.3 900 1000 1100 xx
()
B
2(T
2)
0.325 K
0.934 K
1.552 K
V
g= -0.125 V
0.0 0.1 0.2 0.3 1300 1400 1500 1600 xx()
B
2(T
2)
0.326 K
0.928 K
1.566 K
V
g= -0.145 V
0.0 0.1 0.2 0.3 2000 2200 2400 xx()
B
2(T
2)
0.322 K
0.949 K
1.533 K
V
g= -0.165 V
a
b
c
Figure 5ρxxas a function ofB2forVg=−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.
0 5 10 15 3.8 3.9 4.0 4.1 4.2 ' (0) = 3.79 m2/Vs 0 1 2 3.85 3.90 3.95 ' ( m 2 /V s) T (K ) V g = -0.145 V
' (m
2/Vs)
T (K)
0 5 10 15 4.6 4.7 4.8 4.9 0 1 2 4.60 4.65 4.70 ' (m 2 /V s) T (K)' (m
2/Vs)
T (K)
V g = -0.125 V ' (0) = 4.59 m2/Vs 0 5 10 15 2.9 3.0 3.1 3.2 3.3 ' (0) = 2.89 m2/Vs 0 1 2 2.90 2.92 2.94 T (K) ' (m 2 /V s) V g = -0.165 VT (K)
' (m
2/Vs)
b
a
c
Figure 6 Renormalized mobilityμ′ as a function of T for Vg=−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.
μ0−μ D
ð Þ with μD ≡ μ(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.
At high magnetic fields B > 1/μD, semiclassical effects
should affect the background resistance, resulting in ei-ther 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 ofμDare 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 ofμD/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. Sinceμqcounts all scattering events whereasμDis
sensi-tive only to large-angle ones, we can deduce the pre-dominant 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 transi-tion 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 char-acterized by opposite T dependences and are character-ized byρxx~ ρxy. In his argument,μ is taken to be the
transport mobility. Nevertheless, recent experimental re-sults [11-13] demonstrate that different mobilities should be introduced to understand transport near a dir-ect I-QH transition; the observed dirdir-ect 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-inde-pendent point that corresponds to a direct I-QH transi-tion. Moreover, the strongly T-dependent Hall slope
induced by e-e interactions may affect the position of ρxx =ρxyat 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 ~ρxyeven 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 ap-parent oscillations inρxxaround Bcand that the onset of
strong localization occurs at B > 1.37 T, as characterized by a well-quantizedν = 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
0.0 0.6 1.2 1.8 2.4 3.0 0 400 800 1200 1600 2000 2400 xx
&
xy()
Bc = 0.62 TV
g= -0.05 V
B (T)
H597
0.0 0.6 1.2 1.8 2.4 3.0 0 600 1200 1800 2400 3000H597
B c = 0.68 TB (T)
V
g= -0.1 V
xx&
xy()
a
b
Figure 7ρxxandρxyas functions ofB at various T ranging from 0.3 to 2 K. For (a) Vg=−0.05 V and (b) Vg=−0.1 V.
effective amount of disorder and carrier density in the 2DES. As shown in Figure 2a, by increasing Vgto−0.125
V,ρxxbecomes smaller thanρxyat Bc~ 0.26 T, whileρxx
~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
ap-pears 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 whereρxx~ρxyis governed
by 1/μDand does not always correspond to that
respon-sible for a direct I-QH transition as the influence of e-e interactions is not significant. As a result, ρxx~ ρxycan
occur on both sides of Bcas 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, re-spectively. Therefore, survival of the SdH theory for 0.46 T ≤ B ≤ 1.03 T reveals that semiclassical metal-lic transport may coexist with quantum localization. The superimposed background MR may be the reason
0.6 0.9 1.2 1.5 2 4 6 n = 1.90x1011 cm-2 V g = -0.05 V q = 0.6529 m -2 /Vs 1/ B (T-1) ln( xx /( /sinh ) 0.3 K 0.6 K 1 K 1.5 K 2 K 0.6 0.9 1.2 1.5 3 4 5 6 7 n = 1.46x1015m-2 q = 0.5966 m 2 /Vs 1/ B (1/T) ln( xx /( /sinh ) 0.3 K0.6 K 1 K 1.5 K 2 K V g = -0.1 V 0.7 1.4 2.1 3520 3560 3600 3640 4800 Vg = -0.05 V V g = -0.1 V
R
H(/
T
)
T (K)
a
b
c
Figure 8RHand ln(Δρxx(B, T)/D(B, T)). (a) RHas 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.
0.0 0.7 1.4 2.1 2.56 2.58 2.60 2.62 2.64 2.66 2.68 B c = 0.59 T ' (m 2 /Vs) T (K) V g = 0 V '(0) = 2.65 m2/Vs 0.0 0.7 1.4 2.1 1.78 1.80 1.90 1.92 Bc = 0.66 T T (K) ' (m 2 /Vs) V g = -0.075 V '(0) = 1.90 m2/Vs 0.0 0.7 1.4 2.1 2.04 2.06 2.14 2.16 '(0) = 2.16 m2/Vs V g = -0.05 V T (K) ' (m 2 /Vs) Bc = 0.62 T 0.0 0.7 1.4 2.1 1.46 1.48 1.58 1.60 B c = 0.68 T '(0) = 1.59 m2/Vs V g = -0.1 V T (K) ' (m 2 /Vs)
a
b
c
d
Figure 9μ′ 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.
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 back-ground MR. Similar behavior can also be found for Vg=
−0.145 V even though spin splitting is unresolved, indicat-ing 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 corre-sponding to the onset of strong localization occurs at B = 1.12 T.
In order to check the validity of our present results, fur-ther 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 theν = 1 or ν = 2 QH state near Bc. The
cross-ing ofρxxandρxyis found to occur at B ~ 0.5 T which is
smaller than Bc. As we decrease Vg to−0.1 V, thereby
in-creasing 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 ratioρxy/ρxxat Bc, whose
value is 1.57 and 1.31 for Vg = −0.05 and −0.1 V,
respectively.
The interaction-induced parabolic NMR can be observed at both gate voltages. This result, together with the negli-gible T dependence of the Hall slope as shown in Figure 8a, implies that the ballistic part of the e-e interactions domi-nates as mentioned above. Therefore, by analyzing the
ob-served 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 demon-strated that the background MR can validate the SdH the-ory 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 inρxxat B ~ 1.63 T, which
corresponds to theν = 4 to ν = 2 QH plateau-plateau tran-sition. Therefore, it is reasonable to attribute the
discrep-ancy of μ′ obtained by different methods to the
background MR. However, we can see that the value ofμ′ is underestimated by using the first method, which is differ-ent from that in sample LM4640 with the overestimated re-sult. Our experimental results in conjunction with existing reports [37,45-48] 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 [49-51].
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
en-hancement of background disorder with decreasing Vg
may be the reason for such a discrepancy which can be deduced from the ratioμD/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 backscat-tering. Moreover, the parabolic NMR extending well below 1/μD, as shown in Figure 7, provides another
evi-dence 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
disor-dered 2DES both experimentally and theoretically. Conclusion
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 ofρxxand ρxy is governed by the inverse of
the Drude mobility 1/μD and can occur for B > Bc, B <
Bc, and B ~ Bcwhere Bccorresponds to the direct I-QH
transition as the influence of e-e interactions is not sig-nificant. 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 oc-curs 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 es-sential to eliminate the influence of e-e interactions as much as possible.
Competing interests
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.
Acknowledgment
Author details 1
Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan.2Department of Physics, National Taiwan University, Taipei 106, Taiwan.3Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan.4Institute for Solid State Electronics and Center for Micro- and Nanostructures, Technische Universität Wien, Floragasse 7, 1040, Vienna, Austria.5Department of Electrical Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-1920, USA.
Received: 7 May 2013 Accepted: 26 June 2013 Published: 3 July 2013
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doi:10.1186/1556-276X-8-307
Cite this article as: Lo et al.: Tunable insulator-quantum Hall transition in a weakly interacting two-dimensional electron system. Nanoscale Research Letters 2013 8:307.
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