Insulating state to quantum Hall-like state transition in a spin-orbit-coupled
two-dimensional electron system
Shun-Tsung Lo, Chang-Shun Hsu, Y. M. Lin, S.-D. Lin, C. P. Lee, Sheng-Han Ho, Chiashain Chuang, Yi-Ting Wang, and C.-T. Liang
Citation: Applied Physics Letters 105, 012106 (2014); doi: 10.1063/1.4889847 View online: http://dx.doi.org/10.1063/1.4889847
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/1?ver=pdfcov Published by the AIP Publishing
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Insulating state to quantum Hall-like state transition in a spin-orbit-coupled
two-dimensional electron system
Shun-Tsung Lo,1,a)Chang-Shun Hsu,1,a)Y. M. Lin,2S.-D. Lin,2C. P. Lee,2Sheng-Han Ho,3 Chiashain Chuang,3Yi-Ting Wang,3and C.-T. Liang1,3
1
Graduate Institute of Applied Physics, National Taiwan University, Taipei 10617, Taiwan 2
Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 30010, Taiwan
3
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
(Received 27 May 2014; accepted 23 June 2014; published online 9 July 2014)
We study interference and interactions in an InAs/InAsSb two-dimensional electron system. In such a system, spin-orbit interactions are shown to be strong, which result in weak antilocalization (WAL) and thereby positive magnetoresistance around zero magnetic field. After suppressing WAL by the magnetic field, we demonstrate that classical positive magnetoresistance due to spin-orbit coupling plays a role. With further increasing the magnetic field, the system undergoes a direct insulator-quantum Hall transition. By analyzing the magnetotransport behavior in different field regions, we show that both electron-electron interactions and spin-related effects are essential in understanding the observed direct transition.VC 2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4889847]
The insulator-quantum Hall (I-QH) transition1is a fasci-nating physical phenomenon in the field of two-dimensional (2D) physics as it is related to the evolution of extended states between successive QH states with the magnetic field B. Experimentally, a temperature-independent point in the measured longitudinal resistivity qxxof the 2D system at the
crossing field Bc can be ascribed to evidence for the
insulator-quantum Hall transition. ForB < Bc, the 2D system
behaves as an insulator in the sense that qxx decreases with
increasing temperature T.2 For B > Bc, qxx increases with
increasingT, characteristics of a QH state.
Although in some cases the I-QH transition can be well described by the global phase diagram (GPD) developed by Kivelson, Lee, and Zhang,1a direct transition from an insula-tor to a high Landau-level filling facinsula-tor > 2 QH state which is normally described as the direct I-QH transition appears to be beyond the GPD. Such direct I-QH transitions have been observed in various systems such as SiGe hole gas,2GaAs multiple quantum well devices,3 GaAs two-dimensional electron gases (2DEGs) containing InAs quantum dots,4,5a delta-doped GaAs quantum well with additional modulation doping,6,7 GaN-based 2DEGs grown on sapphire8 and on Si,9 InAs-based 2DEGs,10 and even some conventional GaAs-based 2DEGs,11,12 suggesting that it is a universal effect. It is worth mentioning that the aforementioned results are obtained in spin-degenerate systems in which the spin effect and spin-orbit interactions are insignificant. Therefore, it is a fundamental issue whether the direct I-QH transition in a spin-orbit-coupled 2D system can occur, and it is the purpose of this paper to answer this important question. It is found here that the direct I-QH transition occurs with posi-tive magnetoresistivity background. We show that electron-electron interactions, Zeeman splitting, and spin-orbit cou-pling play a role in describing the observed direct I-QH transition.
We have studied an InAs/InAsSb two-dimensional elec-tron system (2DES). Such a device may find applications in narrow-gap material-based transistors and especially in spin-tronic devices owing to its strong spin-orbit coupling. The sample for this study was grown by a solid-source molecular beam epitaxy (MBE) system on a (001) semi-insulating GaAs substrate. The growth temperature was monitored by an infrared pyrometer. The As2 and Sb2 beams were
con-trolled by cracker cells with needle valves, respectively. After a thermal cleaning at 600C ensured the removal of native oxides on the substrate, a GaAs buffer layer of 100 nm was grown at 580C to obtain a smooth surface. A relaxed metamorphic AlSb buffer layer of 1.3 lm grown at 520C was followed to accommodate the lattice mismatch between the active layers and the GaAs substrate. The active layers, from the bottom to the top, consist of a 4-nm-thick AlAs0.16Sb0.84bottom barrier, a strained InAs0.8Sb0.2/InAs/
InAs0.8Sb0.2 (3 nm/9 nm/3 nm) channel, and a 4-nm-thick
AlAs0.16Sb0.84 layer. A 6-nm-thick AlSb top barrier and a
4-nm-thick highly lattice-mismatched In0.5Al0.5As cap layer
were then grown. The In0.5Al0.5As layer keeps the
underly-ing layers from oxidation. They were grown at 500C except the composite channel, which was grown at 470C. The car-riers in the channel were provided by a Te delta-doped layer at the upper AlAs0.16Sb0.84/AlSb interface. Four-terminal
magnetoresistivities were measured using standard ac phase-sensitive lock-in techniques. The magnetic field is applied perpendicular to the plane of the 2D electron system.
Figure1shows the longitudinal and Hall resistivities qxx
and qxyas a function of magnetic fieldB at various T. There
exists an approximately T-independent point in qxx at
Bc¼ 0.73 T. According to the low-field Hall effect, the
car-rier density is calculated to ben¼ 1.43 1016m2. We note
that at the crossing pointBc, the corresponding Landau level
filling factor is about 80 which is much bigger than 2. Therefore, we have observed a direct I-QH transition, consistent with existing experimental results.2–12 We note
a)
that the lack of change in the Hall slope as well as the non-vanishing qxxsuggest that a true QH state is not formed
nearBc. Therefore, it is more appropriate to state that a direct
transition from an insulating state to a quantum Hall-like state is seen. Figures1(b)and1(c)then show the longitudi-nal and Hall conductivities rxxand rxyas a function of
mag-netic fieldB obtained from qxx(B) and qxy(B) according to
rxx(B)¼ qxx(B)/(qxx(B)2þ qxy(B)2) and rxy(B)¼ qxy(B)/
(qxx(B)2þ qxy(B)2). It is found in Fig. 1(c) that rxy is
T-independent over the whole range of 0 < B < 0.9 T, sug-gesting that the carrier density does not vary with T. For B > 0.1 T, rxxis insensitive to the change inT. However, at
low B < 0.1 T, rxx increases with increasing T as shown in
Fig.1(b). Such results are reminiscent of the quantum cor-rections caused by interference and interactions, both of which can give rise to aT dependence of rxx.
As shown in Fig.1, for 0.1 T <B < 0.75 T, we see positive magnetoresistivity in the sense that qxxincreases with
increas-ing B. In order to further study this effect as well as the observed direct I-QH transition, we have performed detailed low-field magneto-transport measurements. Such results are shown in Fig. 2(a). Positive magnetoresistivity centered at B¼ 0 is clearly observed. These experimental results are com-pelling evidence for weak anti-localization (WAL) due to spin-orbit coupling in two dimensions. As shown in Fig.2(b), the converted magnetoconductivity Drxx(B)¼ rxx(B) rxx(0) can
be well fitted to conventional WAL theory.13,14
Drxxð Þ ¼B e2 2p2h 3 2W 1 2þ B2 B W 1 2þ B1 B 1 2W 1 2þ B3 B ln B2 3=2 B1B31=2 !) ; (1)
withB1,B2, andB3as the fitting parameters, in which WðxÞ
is the digamma function. The determined spin-orbit scatter-ing time is approximately and T-independent. As shown in Fig. 2(c), the extracted phase coherence rate increases with increasingT. These results, together with the data shown in Fig. 1, clearly demonstrate that we have observed a direct I-QH transition in a spin-orbit-coupled two-dimensional electron system.
In the field of the direct I-QH transition, it is important to study important physical quantities such as the classical mobility, quantum mobility, and so on.6,7,11,12 To this end, we have performed magneto-resistivity measurements to higher magnetic fields. Figure 3 shows qxx and qxy up to
B¼ 15 T at various temperatures. Quantum Hall plateaus as well as Shubnikov-de Haas (SdH) oscillations are observed, demonstrating that we do have a two-dimensional system. It is known that the amplitudes of the SdH oscillations can be given by15–17
FIG. 1. (a) Longitudinal and Hall resistivities (qxxand qxy) as a function of
magnetic fieldB at various temperatures T. The inset shows the fit to the data atT¼ 0.86 K using the two-band model from Ref.32. (b) Longitudinal conductivity rxx(>4.8 mS) at variousT. (c) Longitudinal and Hall
conduc-tivities (rxxand rxy) at variousT.
FIG. 2. (a) qxx(B) at various T showing
weak antilocalization. From top to bot-tom: T¼ 0.901, 1.100, 1.420, 1.795, and 1.966 K. (b) Experimental data (in black) and fits to the data using Eq.(1)
(in red). Curves have been vertically offset for clarity. (c) The determined phase coherence rate as a function of temperature.
FIG. 3. (a) qxx(B) and qxy(B) at various T for 0 < B < 15 T. From top to
bot-tom at B¼ 11 T: T ¼ 50, 40, 30, 20, 15, 7, 4, and 2 K. The inset shows rxx(B) and rxy(B) at T¼ 1.966 K together with the fit to rxyusing the Drude
model by Eq.(3). (b) The oscillating amplitude Dqxx(B¼ 8.1 T) as a
func-tion ofT. The curve corresponds to the fit to the LK formula D(B, T). (c) Logarithm of Dqxxdivided byD(B, T) as a function of 1/B. (d) The transport
mobility determined from the fit shown in the inset of (a) as a function ofT.
DqxxðB; TÞ ¼ cexpð–p=lqBÞDðB; TÞ; (2) where lq represents the quantum mobility, DðB; TÞ
¼ 2p2
kBm T=heBsinhð2p2kBm T=heBÞ, and c is a con-stant relevant to the value of qxx at B¼ 0 T. The effective
mass is measured to be 0.050m0from the thermal damping
of SdH oscillation amplitude15–17 at B¼ 8.1 T, where mois
the rest mass of an electron, as shown in Fig.3(b). In addi-tion, the amplitudes of the SdH oscillations at various tem-peratures and magnetic fields allow us to calculate the quantum mobility lqto be 0.085 m
2
/V s, which is presented in Fig.3(c). By locating the oscillation extremes, one is able to obtain the carrier densityn¼ 1.43 1016m2, consistent with the estimation from the low-field Hall measurements. On the other hand, the classical mobility lc is estimated to
be 1.66 m2/Vs by fitting the experimental rxy(B) to the
Drude rxy(B)¼ nel 2
B/(1þ (lB)2) as shown in the inset of Fig.3(a). The obtained lcat variousT is shown in Fig.3(d).
It is observed that lcis independent ofT up to 30 K. A slight
decrease of lcatT > 30 K is due to the influence of
electron-phonon scattering. Since lc/lq¼ 20 which is much greater
than 1, in our system the dominant scattering mechanism can be ascribed to remote ionized impurity scattering.18We note at the crossing point lcBc¼ 1.2–1, consistent with
Huckestein’s argument.19 In contrast, the product lqBc is
about 0.06 which is significantly lower than 1. Our results on the direct I-QH transition obtained in a spin-orbit-coupled 2DES are consistent with those obtained in various conven-tional, spin-degenerate 2D system in the sense that lqBc 1
and lcBc 1.
In most cases, the direct I-QH transition occurs with negative magnetoresistivity background in the sense that around the crossing point qxx decreases with increasing B.
This can be understood within the following picture. Since the direct I-QH transition occurs when weak localization is suppressed at high magnetic fields. In this regime, negative magnetoresistivity which varies quadratically with B due to electron-electron interactions is generally observed. Interestingly, in our case, the observed direct I-QH occurs in the background of positive magnetoresistivity. Therefore, it is important to study this positive magnetoresistivity back-ground. From the high-field magnetotransport data, we can estimate the transport relaxation time s 4.7 1013 s, mean free pathl 3.3 107m, diffusion constantD 0.11 m2/s, and the Fermi wave vector kF 3.0 108m1, which
is required in the following analyses.
In our devicekFl¼ 98, demonstrating that our system is
weakly disordered and hence validating the use of quantum correction theory for analyzing the data.20–25In the ballistic regime (kBTs=h > 1), electron-electron interactions may give rise to positive magnetoresistivity.26,27 Moreover, the classical mobility would be renormalized by this ballistic-type interaction and thereby becomes T-dependent.23,24 However, in our system h=ðksÞ 16 K and therefore for T 16 K the observed positive magnetoresistivity in Fig.1(a) cannot be due to electron-electron interactions in the ballistic regime. In Fig.3(d), we know that the classical mobility lc is independent of T (<30 K). Therefore, our
system is in the diffusive regime (kBTs=h 1). It is
pre-regime contributes to rxx only.
20,21
Since the classical Dude conductivities in the magnetic field have the form rD xxðBÞ ¼ nelc=ð1 þ ðlcBÞ 2 Þ and rD xyðBÞ ¼ nel 2 cB= ð1 þ ðlcBÞ 2
Þ, the magnetoresistivity considering this type of interactions dree xxðT; BÞ is given as qxxð Þ B 1 r0 1 r02 1 l2 cB 2 dree xxðT; BÞ; (3)
by performing matrix inversion of the conductivity tensor, where r0 rDxxðB ¼ 0Þ.
21,22 When
glBB < kBT, where g is
the Landeg-factor, lB is the Bohr magneton, the correction
due to electron-electron interactions would become inde-pendent of B, that is, dree
xxðTÞ ¼ e
2
2p2hKeeln kBhTs
< 0 in Eq. (2), where the prefactorKeeis approximatelyB-independent
here.21,22 Therefore, in GaAs-based 2D systems whose gGaAs 0.44 at low fields,28 quadratic negative
magnetore-sistivity will exist for T > 0.22 K at B around 0.73 T. However, for InAs-based devices whoseg can be as large as 10gGaAs,29Zeeman splitting could play a role at Bc¼ 0.73 T
when T < 2.2 K, which covers the whole T range in our measurements shown in Fig.1(a). We note that according to the theory of interactions in the diffusive regime, Kee will
increase with increasing the Zeeman splitting when glBB > kBT,20which indicates that the resulting
magnetore-sistivity is still negative but deviates from the quadratic form. Therefore, Eq.(3)cannot provide good enough explan-ations for our data. Nevertheless, it should be noted that the theory is derived based on the short-range disorder.30,31 In our case, long-range disorder predominates the system since lc/lq¼ 20. In addition, as will be shown later, a classical
mechanism may lead to positive magnetoresistivity back-ground, which can overwhelm the negative magnetoresistiv-ity induced by interactions described by Eq. (3). We note that Btr 3 mT, within which the localization effect is
strongest, Therefore, the observed T dependence of qxx at
B > 0.13 T Btrshown in Fig.1(a)should still be ascribed
to electron-electron interactions.
Since we have observed weak anti-localization due to spin-orbit coupling, we should consider the two-band model for two spin species. As shown in the inset of Fig. 1(a), our data can be well fitted by the two-band model developed by Zaremba,32 where intersubband scattering is included. Therefore, the observed positive magnetoresistivity back-ground can be ascribed to the two spin-split bands due to spin-orbit interactions. In InAs-based 2D systems, spin-orbit coupling and Zeeman effect are expected to be strong. Further experimental and theoretical work on the magneto-transport with the presence of electron-electron interac-tions33in such a system is highly required.
In conclusion, we present an experimental investigation of direct I-QH transition in a spin-orbit coupled 2D electron system. Instead of negative magnetoresistivity, we show that a direct I-QH transition can occur with positive magnetore-sistivity background. By studying the conductivities at low fields and SdH oscillations at high fields, our results demon-strate that the observed transition in such a system is related to Zeeman splitting, spin-orbit coupling, and
electron-This work was funded by National Taiwan University (Grant No. 103R7552-2), and in part, by the Ministry of Science and Technology, Taiwan. We would like to thank Professor Xi Lin at PKU, China for discussions.
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