Probing Landau quantization with the presence of insulator–quantum Hall transition in a GaAs
two-dimensional electron system
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J. Phys.: Condens. Matter 20 (2008) 295223 (5pp) doi:10.1088/0953-8984/20/29/295223
Probing Landau quantization with the
presence of insulator–quantum Hall
transition in a GaAs two-dimensional
electron system
Kuang Yao Chen
1, Y H Chang
1, C-T Liang
1, N Aoki
2, Y Ochiai
2,
C F Huang
3, Li-Hung Lin
4, K A Cheng
5, H H Cheng
6, H H Lin
7,
Jau-Yang Wu
8and Sheng-Di Lin
81Department of Physics, National Taiwan University, Taipei 106, Taiwan, Republic of China 2Department of Electronics and Mechanical Engineering, Chiba University, Chiba 263, Japan 3National Measurement Laboratory, Centre for Measurement Standards, Industrial
Technology Research Institute, Hsinchu 300, Taiwan, Republic of China
4Department of Applied Physics, National Chiayi University, Chiayi 600, Taiwan, Republic of China
5Department of Electronic Engineering, Lung-Hwa University of Science and Technology, Taoyuan 333, Taiwan, Republic of China
6Centre for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan, Republic of China
7Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan, Republic of China
8Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China
E-mail:[email protected]@faculty.chiba-u.jp Received 12 January 2008, in final form 10 June 2008 Published 1 July 2008
Online atstacks.iop.org/JPhysCM/20/295223 Abstract
Magneto-transport measurements are performed on the two-dimensional electron system (2DES) in an AlGaAs/GaAs heterostructure. By increasing the magnetic field perpendicular to the 2DES, magneto-resistivity oscillations due to Landau quantization can be identified just near the direct insulator–quantum Hall (I–QH) transition. However, different mobilities are obtained from the oscillations and transition point. Our study shows that the direct I–QH transition does not always correspond to the onset of strong localization.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The insulator to quantum Hall (I–QH) transition in a two-dimensional electron system (2DES) at low perpendicular magnetic fields B has attracted much attention [1–11]. Theoretically, the direct I–QH transition from the insulator to an integer QH state ofν = 1 is forbidden in an infinite, non-interacting 2DES with an arbitrary amount of disorder, where
ν is the Landau-level filling factor [1–3]. In such a system, the only allowed state at B = 0 is the insulating one, and the
2DES undergoes the I–QH transition to enter theν = 1 QH state [12,13]. Realistically, however, only systems of finite sizes are available, and the effects of the electron–electron (e– e) interaction are significant in some 2DESs [4,5,14–18]. As a result, the 2DESs may experience the direct I–QH transition from the low-field insulator to QH states of higher filling factors [2–4,8,16–18]. Such a transition can be related to the zero-field metal–insulator transition, in which e–e interaction cannot be ignored [4]. Given that most 2DESs show metallic behaviour at B = 0, the investigation of the direct I–QH
J. Phys.: Condens. Matter 20 (2008) 295223 K Y Chen et al transition at low B should be conducted in low-mobility
2DESs [1,12].
The mechanisms for the direct I–QH transition are still under debate [5–7, 11,17, 18]. Huckestein [5] argued that such a transition is a crossover from weak localization to Landau quantization rather than a phase transition. Therefore the observed transition or crossing point is not a critical point. According to Huckestein’s argument, such a point should occur as the product
μB = 1. (1) Here μ is the mobility such that the strong localization due to high-field Landau quantization becomes important when the product μB, which equals the ratio of Landau-level spacing to broadening, is large enough. To be a measure for Landau quantization, μ should be the quantum mobility. Because the strong localization is believed to be important to the QH liquid, it seems natural that a 2DES undergoes the direct I– QH transition at μB = 1 as we increase the perpendicular magnetic field. However, experimental evidence of quantum phase transition has been observed near the transition point [8]. In addition, the existence of Landau quantization in the low-field insulator indicates that its onset may be irrelevant to such a transition [9, 10]. In fact, Landau quantization could be unimportant to the crossover because its feature is absent near the crossing point in some reports [14, 15]. Corrections based on the e–e interaction [14–16,18,19] and low-field Landau quantization effects [9–11] are discussed in the literature. On the other hand, magneto-oscillations due to Landau quantization appear just near the direct I–QH transition with increasing B in some reports [2, 3, 8]. Huckestein’s argument seems correct if we identify the onset of Landau quantization by the appearance of magneto-oscillations. The observations of
ρxy/ρxx ≈ 1, (2)
near the transition points [2, 3] are also consistent with Huckestein’s argument becauseρxy/ρxx = μB in the Drude
model if the transport and quantum mobilities are the same. Here ρxx and ρxy are the longitudinal and Hall resistivities,
respectively. To understand the direct I–QH transition, therefore, we shall re-examine the 2DESs where Landau quantization induces oscillations just near the transition point occurring as equation (2) becomes valid with increasing B.
In this study, we report a magneto-transport investigation on the 2DES in an AlGaAs/GaAs heterostructure. With increasing magnetic field B, amplitudes of resistivity oscillations ρxx following the Shubnikov–de Haas (SdH)
formula [20–24] ρxx ∝ χ sinhχexp −π μB (3) with χ = 4π3km∗T/heB can be identified just as the 2DES undergoes the direct I–QH transition. Here T is the temperature, k, h, e, and m∗ are denoted as Boltzmann constant, Plank constant, electron charge, and effective mass, respectively. The oscillations are features of Landau quantization, so it seems that the observed direct transition occurs near the onset of Landau quantization just as suggested
Figure 1. A Schematic diagram showing the sample structure.
by Huckestein. In addition, equation (2) is valid at the transition point. However, different mobilities should be introduced just as in references [14,15] becauseμB is much smaller than 1 at the crossing point. One is for the direct I–QH transition and the other is for Landau quantization. Therefore, corrections to Huckestein’s argument should be taken into account even when the onset of Landau quantization can be approximated by the transition point where equation (2) is valid.
The experimental conditions are described in section2, and the investigations on mobilities near the I–QH transition are discussed in section 3. Effects due to electron–electron interaction, electron–phonon scattering and disorder-enhanced electron–electron scattering are mentioned in section4, and the conclusion is made in section5.
2. Experimental details
The sample (LM4646) used in this study is an AlGaAs/GaAs heterostructure. Figure 1 shows its structure, where some Si atoms are doped in the 20 nm-wide GaAs quantum well to serve as the scattering sources. It is known that we can suppress the mobility to probe the integer quantum Hall effect by deliberately introducing some scattering sources in the quantum wells [3, 9, 10]. The sample is made into the Hall pattern with the channel width 80μm by standard optical lithography, and AuGeNi alloy is annealed at 450◦C to fabricate the ohmic contacts. The magneto-transport measurements are performed in a top-loading He3system with the superconducting magnet.
3. Insulator–quantum Hall transition and mobility
analysis
Figure2shows the curves of the longitudinal resistivityρxx(B)
at different temperatures and Hall resistivity ρxy(B) at the
temperature T = 4 K under a low-frequency AC driving current of 40 nA. At low B, the 2DES behaves as an insulator such thatρxx increases with decreasing T . The insulator is
terminated at B = 3.5 T ≡ Bc, and ρxx decreases with
decreasing T at B> Bc. Therefore, Bcis the transition point. The filling factor ν ∼ 8 at Bc, and oscillations periodic in 1/B are observed when the sample behaves as a QH liquid at B > Bc. From the oscillating period in 1/B, the carrier 2
Figure 2. Longitudinal and Hall resistivity as a function of magnetic
field (B) at various temperatures T . The dotted line indicates the transition point Bc. The inset shows ln(ρx x/(χ/sinhχ)) as a function of 1/B at T = 0.7, 0.9, 1.1, 1.3, 1.5, 3 and 4 K, respectively.
concentration n = 6.8 × 1015 m−2. We can see in figure2, that an SdH dip appears as B ∼ Bc, so the observed I–QH transition at Bcis a direct one [2,3,5]. In figure2, magneto-oscillations cannot be observed at low B until we increase the magnetic field to about B = Bc. Since such oscillations are due to Landau quantization, the 2DES provides us an opportunity to probe the direct I–QH transition which occurs as Landau quantization can just be identified. In addition, we can see thatρxx = 3.4 k ≈ ρxy = 3.1 k = neB at Bc
at T = 4 K although the Hall slope is weakly T -dependent. So the observed transition occurs as ρxx/ρxy ≈ 1, which
seems to be consistent with Huckestein’s argument. The low-field oscillations are expected to follow equation (3), the SdH formula. To analyze the mobility from equation (3), we note that ln(ρxx/(χ/sinhχ)) = const−π/(μB). We can see from
the inset to figure2that the data of ln(ρxx/(χ/sinhχ))−1/B
at different temperatures collapse well into a single straight line when we take m∗ = 0.067m0 as the expected value in a GaAs 2DES. From the slope of ln(ρxx/(χ/sinhχ)) − 1/B,
the quantum mobility μ = 0.13 m2 V−1s−1. Therefore, we can obtain the product μB = 0.46 at the transition point B = Bc. Such a product deviates much from 1, and thus our result is inconsistent with Huckestein’s argument although the direct I–QH transition occurs just as the magneto-oscillations due to Landau quantization can be observed under equation (2). Since the conventional SdH formula is based on Landau quantization without considering strong localization, the fact that the experimental data shown in the inset to figure 2 can be well fitted to equation (3) suggests for
B 5.4 T strong localization may not be significant in our
system.
It is known that Landau quantization can result in magneto-oscillations as the productμB < 1 [25]. Therefore, the appearance of magneto-oscillations near Bc does not indicate that the transition occurs just as equation (1) is valid. While numerical studies show that such transitions can occur just as μB ≈ 1 in a non-interacting 2DES, Landau quantization can induce magneto-oscillations at μB < 1
-2 -4 In [ Δρxx /( χ /sinh χ )] -2 -4 In [ Δρxx /( χ /sinh χ )] 0.6 0.9 1.2 1/B (1/T) 0.9 1.2 1/B (1/T) (b) (a)
Figure 3. ln(ρx x/(χ/sinhχ)) as a function of 1/B at
(a) Vg= +0.15 V and (b) Vg= 0 at different temperatures T .
where such a 2DES is an insulator [11]. The coexistence of magneto-oscillations and insulating behaviour can be explained by the percolation theory [26, 27]. We note that Huckestein considered only a single mobility based on the Drude model, but another mobility μ has been introduced in [14–16,18]. The mobility μ corresponds to the quantum mobility while μ can be related to the transport mobility although renormalization effects may be important [16]. The direct I–QH transition should occur asμ B = 1, and thus we
obtainμ = 1/Bc = 0.29 m2 V−1s−1 ≈ 2.2μ. Therefore, different mobilities should still be taken into account even as Landau quantization can be identified near Bc with increasing B.
To further check Landau quantization near direct I–QH transitions, we re-examine the data published in our previous report [8]. In that report, we also investigated direct I–QH transitions, near which magneto-oscillations can be identified, at low magnetic fields in a gated 2DES. Magneto-oscillations can be observed as the filling factorν ∼ 10 in such a 2DES when the gate voltage Vg = +0.15 and 0 V, and we can apply equation (3) to analyze the quantum mobility after the appearance of I–QH transitions. Figures3(a) and (b) show the curves of ln(ρxx/(χ/sinhχ)) − 1/B at these two gate
voltages, and the slopes yieldμ = 0.53 and 0.47 m2V−1s−1 at Vg = +0.15 and 0 V, respectively. On the other hand, the transition points yield μ = 1.9 and 1.7 m2V−1s−1 under these two gate voltages. The quantum mobilityμ is much lower than the mobilityμ obtained from the transition point.
J. Phys.: Condens. Matter 20 (2008) 295223 K Y Chen et al
Figure 4. Hall slope as a function of T . The black squares represents
the result obtained at I = 40 nA while the open square corresponds to the data obtained at I= 12 nA. The straight line corresponds to the best linear fit at T = 0.5–4 K. The lower inset shows the logarithm of electron effective temperature Teas a function of logarithm of current I determined from the zero-field resistivityρx x at different lattice temperature T . The best linear fit corresponds to
Te∝ Iαwithα = 0.46. The upper inset shows the inverse of phase coherence time 1/τφas a function of T .
Therefore, different mobilities should also be introduced to understand the direct I–QH transitions.
In Huckestein’s argument, the direct I–QH transition separates the weak-localization regime from the QH liquid due to the strong localization under Landau quantization. At low B, however, either Landau quantization or the quantum Hall effect can be irrelevant to the strong localization effect [20,23,24,28–30]. The onset of magneto-oscillations following equation (3) near the transition field Bc, in fact, does not indicate the importance of the strong localization to the direct I–QH transition because equation (3) can hold without any localization effect [20,31]. Huckestein’s argument is valid only if the onsets of both the strong localization and Landau quantization are atμB ≈ μ B ≈ ρxy/ρxx ≈ 1. Our study
shows that the direct I–QH transition does not always indicate the onset of strong localization even when Landau quantization can be identified near the transition point with increasing B.
4. Discussion
Corrections based on the e–e interaction effect [14–16] have been taken into account for the direct I–QH transition when the magnetic field is too weak to induce the high-field strong localization effect. In our study, as shown later, there exists evidence for e–e interaction and scattering although semiclassical and electron–phonon effects should be also considered.
4.1.T -dependent Hall slope and e–e interaction
The e–e interaction effect can modify the 2D density of states near the Fermi level, giving rise to a logarithmic T -dependent Hall slope of a 2DES [32]. As shown in figure4, the Hall slope is logarithmic T -dependent at T = 0.5–4 K in the 2DES in
sample LM4646. Since the carrier density determined from the oscillations inρxx remains constant over the same temperature
range, the observed logarithmic T -dependent Hall slope can only be ascribed to e–e interaction effect within our system. This experimental evidence for e–e interactions suggests that such effects could be important to the observed I–QH transition in our system. The parabolic negative magneto-resistance, however, is not apparent at μB < 1 in figure2 although it is also expected under the e–e corrections [14]. In addition, we note that the magneto-oscillations are absent at Bcin [14] and [15] while they appear near the transition point in our study and in [2,3]. In different 2D systems, therefore, it is possible that the dominant effects and/or parameters are not the same at low fields [14,15,33].
4.2. Electron effective temperature
We can see from figure4 that the Hall slope under a current
I = 40 nA deviates somewhat from the expected logarithmic T dependence at the lowest temperature. To understand the
mechanism for the deviation, we note that ρxx at B = 0
is I -dependent with increasing current. Here ρxx(B = 0)
represents the value of ρxx at zero magnetic field. The I
-dependence indicates the existence of the current heating, under which the electron effective temperature Te is higher than the lattice temperature T [34]. Therefore, effects due to electron–phonon interaction could be important in our study for electrons to transfer the extra energy to the lattice, which can induce the deviation of the Hall slope at low T . The temperature dependence of ρxx at B = 0 can be used as a
self thermometer to determine Te as follows. Under a low-current without inducing electron heating, Teshould equal the lattice temperature T and the dependence ofρxxat B = 0 with
respect to Te = T can be obtained by direct measurements. Because ρxx at zero magnetic field is a decreasing function
of T (or Te) under a low enough current in our study, the value of ρxx and Te is in one–one correspondence at B =
0. Then at a fixed lattice temperature T , we can raise Te by increasing the current I and determine Te from such a correspondence. In this way, the I -dependence of Teat zero field is determined, and the lower inset shows the relation between Teand I at different lattice temperature when B= 0. We can see from such an inset that the zero-field resistivity data shows Te ∝ Iα with the exponent α = 0.46 ≈ 0.5, which is expected under the electron–phonon interaction [35]. The current and temperature dependences of the Hall slope yields α = 0.53, which is also close to 0.5. Actually the low-field regime is unstable in the global phase diagram of the quantum Hall effect [1], and more studies are necessary to clarify the dominant effects and/or parameters at low magnetic fields [11,14–16,18,19,21–24,28–30].
4.3. Phase coherence time analysis and e–e scattering
By decreasing the current to I = 12 nA, as indicated by the open square in figure4, the deviation on the logarithmic T -dependence of the Hall slope at low T can be removed. In addition, we note that the direct I–QH transition atμB = 1 can still be related to the e–e interaction effect when corrections 4
to the negative magneto-resistance are taken into account. Moreover, the linear T -dependence of the inverse of the phase coherence timeτφ in the upper inset to figure4indicates the scattering due to the e–e interaction while the nonzero intercept shows the zero-temperature dephasing [37]. The slope of 1/τφ–T equals 3.45 × 1010 s−1 K−1, which is a reasonable value under the e–e scattering [32]. The phase coherence time
τφis obtained by fitting our data to the low-field equation [36]
σxx(B) = −e 2 πh 1 2 + B0 B − 1 2+ Bφ B , (4)
where is the digamma function and B0and Bφcorrespond to transport and phase coherence rates, respectively [32]. Therefore, the direct I–QH transition in our study could be dominated by the e–e interaction effect rather than the onset of Landau quantization although different mechanisms should be introduced to understand the details. In our study, bothμ and μ remain the same after decreasing the driving current, which also indicates that the current heating and/or electron– phonon interaction is irrelevant to the difference between these two mobilities.
5. Conclusion
In conclusion, we investigate Landau quantization and the direct I–QH transition in the two-dimensional electron system in an AlGaAs/GaAs heterostructure. Our study shows that such a transition does not occur as μB = 1 even when Landau quantization can be identified near the transition point by the appearance of magneto-oscillations as ρxy/ρxx ≈ 1.
Therefore, our study supports that different mobilities should be introduced for the direct I–QH transition and Landau quantization. The temperature dependences of the Hall slope and dephasing time indicate the importance of the effects of the e–e interaction to the direct I–QH transition although different mechanisms should be considered for the details of such a transition. The appearance of Landau quantization or direct I– QH transition, in fact, does not always correspond to the onset of the strong localization effect giving rise to quantum Hall liquids.
Acknowledgments
This work was funded by the NSC, Taiwan. We would like to thank Yu-Ru Li, Po-Tsun Lin, Yen Shung Tseng and Chun-Kai Yang for their experimental help. KYC gratefully acknowledges financial support from Interchange Association, Japan (IAJ) and the NSC, Taiwan for providing a Japan/Taiwan Summer Program student grant. We would like to thank Professor Efrat Shimshoni and Dr Marian Nita for helpful discussions.
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