Landau quantization, Localization, and Insulator-quantum. Hall Transition at Low Magnetic Fields

Transcription

1 Landau quantization, Localization, and Insulator-quantum Hall Transition at Low Magnetic Fields Tsai-Yu Huang a, C.-T. Liang a, Gil-Ho Kim b, C.F. Huang c, C.P. Huang a and D.A. Ritchie d a Department of Physics, National Taiwan University, Taipei 106, Taiwan b Department of Electronic and Electrical Engineering, Sungkyunkwan University, Suwon , Korea c National Measurement Laboratory, Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu~300,~Taiwan d Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom We have studied the localization, Landau quantization, and insulator-quantum Hall (I-QH) transition in a gated GaAs two-dimensional electron system (2DES) containing self-assembled InAs quantum dots. Shubnikov-de Haas oscillations arising from Landau quantization are observed in the low-field insulating phase, and with increasing the magnetic field B the 2DES undergoes a direct I-QH transition to enter the quantum Hall state of the filling factor ν=4. The direct I-QH transition, in fact, does occur as B approaches the inverse of the mobility µ while the Hall and longitudinal resistivities may be different at its transition point. At low B, in fact, Landau quantization can modulate the density of states without resulting in the formation of quantum Hall states.

2 In the presence of a strong perpendicular magnetic field B, the Landau quantization becomes important, causing the formation of Landau levels in a two-dimensional (2D) system. [1-19] It is well established that Landau quantization can strongly modify the electrical properties of a 2D system. With increasing B, usually Landau quantization may result in Shubnikov-de Haas (SdH) oscillations [1-3] at low fields and the integer quantum Hall effect (IQHE) [1,4] at high fields. It was shown that in the longitudinal resistivity ρ xx the amplitude ρ xx of SdH oscillations can be approximated by [1] ρ xx (B,T)=4ρ 0 exp(-π/µb)d(b,t), (1) with D(B,T)=2π 2 k B m*t/ ebsinh(2π 2 k B m*t/eb ). Here µ, m *, B, T, k B, and are the mobility, effective mass, magnetic field, temperature, Boltzmann constant, and Plank constant divided by 2π, and the parameter ρ 0 is expected to be the value of ρ xx at zero magnetic fields. From SdH oscillations, therefore, we can obtain the effective mass and mobility, both of which are important physical quantities of a 2D system. On the other hand, in the IQHE localization effects are important and one is able to study magnetic-field-induced transitions, [5-8] which are good examples of quantum phase transitions. In the global phase diagram (GPD) [5] of the IQHE, all magnetic-field-induced transitions are regarded equivalent though they are divided into two types, plateau-plateau transitions and insulator-quantum Hall (I-QH) transitions. [6,8] Recently there has been a great deal of interest in the low-field IQHE. [9-19] For the high-field IQHE, the universality on critical conductivities has been observed. [6] However, it has been shown that such a universality might not exist at low fields. [9] It is widely believed that at B=0, all 2D systems are insulating. [5,10,11] With increasing B, a low-mobility 2D system can undergo an I-QH transition. [10-21] In the GPD, an I-QH transition is only allowed between the insulating phase and a filling

3 factor ν=1 quantum Hall (QH) state. Therefore it is an interesting but unsettled issue whether the direct I-QH transitions from the low-field insulator to a high filling factor (ν 3) are genuine phase transitions. [11-17] Experimental [15-17] and theoretical studies [12-14] show that such transitions are phase transitions, and the universality [14,15] ρ xy ~ρ xx (2) is expected at their transition points, where ρ xy is the transverse resistivity. In the Drude model ρ xy /ρ xx =µb and hence from Eq. (2) we have [11,15,17] µb~1, (3) in which the product µb just equals the ratio of the spacing to linewidth in Landau quantization. Therefore, well-separated Landau bands are important to the direct I-QH transitions, which is natural because with increasing B a 2DES becomes the QH liquid arising from Landau quantization after passing through such transitions. [15,17] Although Huckestein [11] argued that the direct I-QH transitions are only crossovers rather than phase transitions, in his arguments Eq. (2) should still hold at the transitions. However, there are reports on the fail of Eq. (2) in the direct I-QH transitions. [17,18] It has been shown that the crossover or transition from the low-field insulator to Landau quantization actually may cover a wide range in B rather than a small region near the I-QH transitions. [18,20] Motivated by these recent results on IQHE at low fields, we study a gated 2D GaAs electron system containing self-assembled InAs quantum dots. Depositing self-assembled InAs quantum dots in the GaAs quantum well has proved to be a reliable way to introduce scattering in the 2DEG [19]. The strain fields due to the self-assembled InAs quantum dots cause strong scattering experienced by the 2DES, providing the necessary disorder to observe I-QH transitions [19]. In this Letter, we report experimental results on localization, Landau quantization and

4 insulator-quantum Hall transition at low magnetic fields. From our study, the low-field Landau quantization is governed by the semiclassical SdH theory such that it modulates the density of states without inducing the QH liquid. The direct I-QH transitions, in fact, occurs as µb~1 although the expected universality Eq. (2) may not exist. For convenience, in the following we denote 0 ν transitions as the I-QH transitions separating the insulator from the quantum Hall state of the filling factor ν, where the number 0 presents the insulating region. The structure of the sample used in this study has been shown in Ref. 18. The sample is made into Hall pattern by the standard lithography and etching processes and a NiCr/Au gate was evaporated on the surface. By changing the applied gate voltage V g, we can vary both the carrier density and mobility of the sample. Magneto transport measurements were performed with a top-loading He 3 system at temperatures ranging from 0.25 to 1.6 K in a superconductor magnet. A phase sensitive four-terminal ac lock-in technique was used with a current of 10 na. Figure 1 shows the low-field curves of ρ xx at T= 0.25 K, 0.47 K, and 0.61 K at V g =-0.06 V. At low B the sample behaves as an insulator in the sense that ρ xx decreases as T increases. With increasing B SdH oscillations due to Landau quantization appears when B>0.4 T, and ρ xx becomes T-independent at B=0.94 T B c. From the oscillations the carrier concentration n= /cm 2. The T-dependences of ρ xx are different on both sides of B c and QH states of ν=2 and 4 are observed when B>B c, so the 2DES undergoes a 0-4 transition at B c. As shown in the lower inset of Fig. 1, the transition point is T-independent within the experimental errors. However, we can see in the upper inset of Fig. 1 that at T=0.25 K Eq. (2) holds at B=0.61 T B a rather than at the transition point B c, at which ρ xy /ρ xx =1.82. In addition, in our study Landau quantization not only induces QH liquid when B>B c, but also SdH oscillations when B<B c. [18-22] Hence Landau quantization appears in the low-field insulator

5 rather than after passing through the I-QH transition as B increases. While ρ xx (B) decreases as T decreases in QH states because of the high-field localization, [11] we can see in Fig. 1 that ρ xx (B) in the low-field insulator increases as T decreases even at the minima in SdH oscillations. Therefore, with increasing B the high-field localization leading to the QH liquid appears after passing through the I-QH transition rather than at the onset of Landau quantization in the low-field insulating region. To further understand the oscillations in the low-field insulator, we analyzed their amplitudes ρ xx. At the lowest temperature T=0.25 K, as shown in the inset of Fig. 2, in the low-field insulator ln ρ xx is linear with respect to 1/B, which is consistent with Eq. (1) under which ρ xx ~4ρ 0 exp( π/µb) when T is so small that D(B,T) 1. Taking ρ (0) xx(b) as ρ xx at T=0.25 K, we found that the ratio ρ xx (B,T)/ ρ (0) xx(b) with T>0.25 K tend to decreases as B increases, which is also consistent with Eq. (1) of which the T-dependent factor D(B,T) is a function of T/B. We note that Eq. (1) is derived by considering effects of magnetic fields on the density of states (DOS). [1] The appearance of SdH oscillations in the low-field insulator, therefore, indicates that Landau quantization modulates the DOS before the 2DES enters the QH liquid as B increases. Equation (1), in fact, can be obtained from a semiclassical approach. [1] Although the observed insulator in our study is due to the low-field localization and the semiclassical approach may not be sufficient, such a localization effect is suppressed with increasing B. [10,11,23] On the other hand, in Eq. (1) the oscillating amplitude increases rapidly as B increases and such an equation may be still a good approximation for low-field oscillations. As mentioned above, we find that the field and temperature dependences of such oscillations are consistent with Eq. (1), and the effective mass m*=( ) m 0 close to the expected value m 0. As shown in Fig. 2, there is a good fit to ρ xx /D(B,T)=4ρ 0 exp(-π/µb) which yields a mobility µ

6 of cm 2 /V-s. Checking the value of µ at T=0.25 K and 0.32 K, we note that the errors on the effective mass are unimportant to the analysis on µ because ρ xx tends to saturate at low temperatures. At the transition point B c, in fact, the product µb=0.97~1 and Eq. (3) holds although Eq. (2) is invalid in our study. "Decreasing the gate voltage the product \mu B at the0-4 transition can be reduced to 0.87 as Vg=-0.07 V, but this transition and \nu=4 QH state tend to disappear when Vg= V so that we cannot further suppress such a product. From our study, therefore, Eq. (3) is important to the appearance of the direct I-QH transitions and \nu 3 quantum Hall states even when Eq. (2) is invalid at the transition points. Since Eq. (2) is equivalent to Eq. (3) within the Drude model, the reason that Eq. (2) is invalid at B c should be due to that such a model is not applicable to our system. In the Drude model, ρ xx at low B is constant and the mobility µ in Eq. (2) is the scattering mobility. [2,15] However, in our study ρ xx decreases significantly as B increases, and it is known that the mobility relative to the widths of Landau bands is the quantum mobility. While in the standard SdH theory the non-oscillating background in ρ xx should maintain at the constant ρ 0 in Eq. (1), [2] we found that ρ 0 in Eq. (1) is smaller than ρ xx (B,T) when B<B c. The details of the nonoscillating background and the change of ρ 0 in Eq. (1) at different gate voltages will be reported elsewhere. The reason that the Drude model is not applicable could be due to the localization [11,20], two-body interaction [24], or modulation of the density of states in Landau quantization [1]. We note that the appearance of the QH liquid under Eq. (3) does not indicates that QH states should be observed at B~1/µ in a clean sample where the disorder may be so weak that localization leading QH states appear at higher magnetic fields. [1] In addition, Landau quantization appears as µb~1/2 in our study and such a product still provides a rough estimation for its onset. Therefore, good pictures for I-QH

7 transitions could still be obtained under the assumption that at B=1/µ corrections to the Drude model are so weak that Landau quantization appears with Eq. (2). On the other hand, as shown in the inset of Fig. 3, the observations of SdH oscillations in the insulator [18-22] can be due to the modulation of the DOS arising from Landau quantization. In the insulator delocalized states must be above the Fermi energy E F, so all states below E F are localized. At V g = V, as shown in Fig. 3, the 2DES behaves as an insulator even at ν~2 and hence the corresponding QH state is destroyed. The destruction such a QH state, which is the lowest and the most robust one under the unresolved spin-splitting in our study, indicates that the 2DES is an insulator at all B. [5,10,12,16,19] In such a case, we can still observe SdH oscillations in Fig. 3. In conclusion, we performed a magnetotransport study on a two-dimensional gated GaAs electron system containing self-assembled InAs quantum dots. We observed SdH oscillations induced by Landau quantization in the low-field insulator, and the 2DES undergoes a 0-4 direct transition as the magnetic field increases. The low-field Landau quantization, in fact, is governed by the SdH theory and can modulate the density of states without inducing the quantum Hall liquid. While the expected property ρ xy ~ρ xx may be invalid at direct I-QH transitions, we find that such transitions does occur as the product µb~1 and hence well-separated Landau bands are still important. This work was funded by the NSC, Taiwan. The work at Cambridge was supported by the EPSRC, UK. G.H.K. acknowledges financial support from R&D Project for Nano Science and Technology (Contract No. M ) of MOST.

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9 [24] L. Li et al., Phys. Rev. Lett. 90, (2003). Figure Captions: Figure 1. Traces of ρ xx (B) at Vg=-0.06 V at different temperatures. From top to bottom: T= 0.25 K, 0.47 K, and 0.61 K. Inset (a) shows a zoom-in of ρ xx (B) near the critical field B c at various temperatures. From top to bottom: T=0.25, 0.32, 0.47, 0.61, 0.78, 0.93, 1.01, 1.22 and 1.37 K. Inset (b) shows ρ xx (B) and ρ xy (B) at T= 0.25 K. B a corresponds to the onset of SdH oscillations. Figure 2. ρ xx /D(B,T) as a function of the inverse of magnetic field (1/B) at nine different temperatures. The dotted line corresponds to a fit to ρ xx /D(B,T)=4ρ 0 exp(-π/µb). The inset shows ln ρ xx as a function of 1/B at. The dotted line corresponds to a fit to ρ xx (B,T)=4ρ 0 exp(-π/µb) Figure 3. ρ xx (B) at Vg= V at various temperatures. The inset illustrates a schematic diagram showing Landau quantization-modulated density of states (DOS) as a function of energy E. The shaded region could be either extended states or localized states.

10 Fig. 1 6 ρ xy ρ xx & ρ xy (kω) 10 5 B a B c ρ xx B (T) ρ xx (kω) 4 B c ρ xx (kω) B c B (T) B (T)

11 Fig. 2 ρ xx /D (B,T) ln ρ xx /B /B

12 Fig. 3 ρ xx (kω) DOS E F E T = 0.25 K T = 0.50 K T = 0.81 K 20 ν=4 ν=2 1 2 B (T)