An experimental study on (2) modular symmetry in the quantum Hall system with a small spin
splitting
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J. Phys.: Condens. Matter 19 (2007) 026205 (8pp) doi:10.1088/0953-8984/19/2/026205
An experimental study on
(2) modular symmetry in
the quantum Hall system with a small spin splitting
C F Huang1,2, Y H Chang1,7, H H Cheng3, Z P Yang3, H D Yeh1,2,C H Hsu1, C-T Liang1, D R Hang4,5and H H Lin6
1Department of Physics, National Taiwan University, Taipei 106, Taiwan, Republic of China 2National Measurement Laboratory, Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu 300, Taiwan, Republic of China
3Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan, Republic of China
4Department of Materials Science and Optoelectronic Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, Republic of China
5Center for Nanoscience and Nanotechnology, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, Republic of China
6Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan, Republic of China
E-mail:[email protected]
Received 10 July 2006, in final form 27 October 2006 Published 15 December 2006
Online atstacks.iop.org/JPhysCM/19/026205 Abstract
Magnetic-field-induced phase transitions were studied with a two-dimensional electron AlGaAs/GaAs system. The temperature-driven flow diagram shows features of the(2) modular symmetry, which includes distorted flowlines and a shifted critical point. The deviation of the critical conductivities is attributed to a small but resolved spin splitting, which reduces the symmetry in Landau quantization (Dolan 2000 Phys. Rev. B 62 10278). Universal scaling is found under the reduction of the modular symmetry. It is also shown that the Hall conductivity can still be governed by the scaling law when the semicircle law and the scaling on the longitudinal conductivity are invalid.
Magnetic-field-induced phase transitions in two-dimensional electron systems (2DESs) have been an active research topic since the discovery of the quantum Hall effect [1–18]. The law of corresponding states proposed by Kivelson, Lee, and Zhang (KLZ) [2], which was based on the effective-field Maxwell–Chern–Simon theory, provides a powerful method for classifying quantum Hall states and the transitions between them. According to the law of corresponding states, all the magnetic-field-induced phase transitions are of an equivalent class. In the integer quantum Hall effect (IQHE), the equivalence is established by the Landau-level
J. Phys.: Condens. Matter 19 (2007) 026205 C F Huang et al
addition transformation [1,2]. Magnetic-field-induced phase transitions are believed to be good examples of quantum phase transitions [1,19]. Universal properties such as the reflection symmetry [3], universality of critical conductivities [1,4], and universal scaling with the same critical exponent [5,6] are expected and, in addition, it is also expected that the temperature-driven flow lines [1,7,8] are governed by the semicircle law.
Because of the existence of the law of corresponding states, the phase diagram of the QHE has a symmetry equivalent to the0(2) symmetry group, which is a subgroup of the
modular group [8–10,20–24]. The universal properties mentioned above can be taken as the manifestations of the0(2) modular symmetry [8–10]. However, this symmetry relies on the
assumption that all the Landau bands are equally spaced in energy, a condition satisfied in the original paper by KLZ, in which the electrons are assumed to be spinless particles. In the presence of a magnetic field, the Zeeman splitting between the spin-up and spin-down electrons with the same Landau level index is given byE = g∗μBB, where g∗is the effective g factor,
μBis the Bohr magneton, and B is the magnetic field [25–27]. This energy is usually different
from the cyclotron energy of an electron with an effective mass m∗ in a magnetic field B,
E = heB/m∗. Therefore, including spin in the problem, the Landau bands become pairs of Landau bands and are no longer all equally spaced. Dolan [9] pointed out that the law of corresponding states proposed by KLZ is suitable when all the Landau bands are well separated and there is no coupling between them. When the spin splitting is small but is resolved, the two Landau bands with the same cyclotron energy are separated only by a small spin gap. The coupling between the two spin states reduces the modular symmetry from0(2) to (2) [9,10].
With the reduction in symmetry, the particle–hole and Landau-level addition transformations should be modified to [9]
σxx(2 − ν) ↔ σxx(ν), σxy(2 − ν) ↔ 2e2/h − σxy(ν) (1)
σxx(ν + 2) ↔ σxx(ν), σxy(ν + 2) ↔ σxy(ν) + 2e2/h. (2)
Here ν is the filling factor, and σxx and σxy are the longitudinal and Hall conductivities,
respectively. The main difference between the0(2) and the (2) symmetry is that although
derivation of the semicircle law from the duality is still valid, the critical point in the crossover between two Hall plateaus or between quantum Hall and insulating states is no longer fixed by the duality. In the0(2) group, the critical point is fixed at the centre of the semicircle in the
complexσ plane, but the critical point in the (2) group can be at any point on the semicircle. In addition, the reduction of the symmetry also results in the distortion of the temperature-driven flow diagram.
In this paper, we report a magnetotransport study on the gated 2DES in an AlGaAs/GaAs heterostructure. We benefit from the fact that this 2DES has been studied before and is well characterized [18]. In this study, a temperature-driven flow diagram with the features due to
(2) symmetry is constructed by investigating a P–P transition between two states separated by
small spin splitting. Universal scaling is observed under the reduced symmetry, and it is found that the scaling onσxyis more robust than that onσxx. On the other hand, the feature of0(2)
symmetry is observed in the low-field P–P transition where the spin splitting is unresolved and the transition is between Landau levels with different indices. The sample used in this study was grown by molecular beam epitaxy. The 10× GaAs/AlAs (1 × 1 nm) layers, an undoped GaAs layer (500 nm), an undoped Al0.22Ga0.78 layer as spacer (20 nm), a 40 nm
Si-doped Al0.22Ga0.78As layer with the doping concentration 8× 1017cm−3, and a 10 nm
Si-doped GaAs cap layer with the doping concentration 4× 1017 cm−3 were grown on top of
a semi-insulating GaAs substrate. The sample was made into a Hall pattern of 1 mm width with voltage probe spaced 2 mm apart by standard photolithography. Indium was alloyed into the contact region at 450◦C in N2atmosphere to form ohmic contacts, and Al was deposited 2
Figure 1. The longitudinal and Hall resistivitiesρx xandρx yat the temperature T = 0.94 K when
the magnetic field B= 1.5–12 T and the gate voltage Vg= −0.1 V.
onto the surface of the Hall bar to form a Schottky gate. Magneto-transport measurements were performed with a top-loading He3 system and a 0–15 T superconductor magnet. A low-frequency AC lock-in technique was used and a current of 0.1 μA was applied to the sample. From Shubnikov–de Haas oscillations and Hall measurement, the carrier concentration n is determined to be 2× 1011 cm−2 at Vg = −0.1 V. The low-field longitudinal resistivity ρxx
at such a gate voltage increases weakly as the temperature T decreases, and the electron mobilityμ = 7.6 × 103cm2V−1s−1is obtained fromρ
xx = 1/neμ at zero magnetic field at T = 0.31 K. The mobility is so low that no fractional quantum Hall effect is observed in our
study. Therefore, we can focus on the transitions in the IQHE.
Figure 1 shows the curves of the longitudinal and Hall resistivities ρxx and ρxy at
temperature T = 0.94 K with B = 1.5–12 T and gate voltage Vg = −0.1 V. We can see in
figure1that, with increasing B, plateaus of filling factorν = 4, 2, and 1 appear successively. Between two adjacent QH plateaus, the 2DES undergoes P–P transitions. Since there is no QH state of the odd filling factorν = 3 between ν = 4 and ν = 2 QH states, spin splitting is unresolved in the transition separating theν = 4 and 2 QH states. Such a transition is a spin-degenerate P–P transition. On the other hand, theν = 1 QH state due to the spin gap appears at higher B, and the P–P transition between theν = 2 and 1 QH states is a transition between two closely spaced spin states. As reported in [18], the sample undergoes an insulator–quantum Hall (I–QH) transition at B = 14.7 T to leave the ν = 1 QH state and enter the insulating phase. For convenience, we denote such an I–QH transition as a 1–0 transition, where the number ‘0’ presents the insulating phase. In the following, we denote the P–P transition by the filling factors of its adjacent QH plateaus. Thus we observed 4–2, 2–1, and 1–0 transitions with increasing B.
Figure2shows the curves of the conductivitiesσxx andσxyobtained fromρxx andρxyby
σxx(xy)= ρxx(xy)/(ρ2xx+ ρ
2
xy) (3)
in the 2–1 transition. At B = 5.49 T, both σxx andσxy are T -independent and this is the
critical magnetic field, B2−1, for the 2–1 transition. The experimental flow diagram for such
a transition is shown in figure3, in which each solid line corresponds to the T -driven flow line at a magnetic field and the dash–dot line is the traceσxx(σxy) at T = 0.31 K, the lowest
temperature in our study. In the conventional flow diagram for the0(2) group, the transition
point is expected to be located at the centre of the semicircle, and the T -driven flow lines flow in different directions for B> B2−1and B< B2−1. In addition, the flow lines are symmetrical
J. Phys.: Condens. Matter 19 (2007) 026205 C F Huang et al
Figure 2. Traces of the longitudinal and Hall conductivitiesσx xandσx yin the 2–1 transition. The
dash–dot–dot line labels the critical point at the field B2−1= 5.49 T.
Figure 3. The temperature-driven flow diagram for the 2–1 transition. The dash–dot line is the
curveσx x(σx y) at T = 0.31 K and each solid line corresponds to a temperature-driven flow line
at a magnetic field. The symbols, ♦, , and
◦
are for the points at T = 0.94, 0.68, 0.49, and 0.31 K, respectively. The point Q corresponds to the critical point of the 2–1 transition. We can see that the flow line starting at point A does not move towards the point(2e2/h, 0) as T decreases. At the left-hand side of point Q, there are flow lines away from the semicircle at higher temperatures, e.g., the flow line starting from the point C. The dash–dot line and the solid line in the inset are the curveσx x(σx y) at T = 0.31 K and the expected semicircle, respectively.with respect to the vertical line passing through the centre of the semicircle. However, we can see in figure3that, although the semicircle relation is still valid, as can be seen from the nice semicircle that satisfies the relation
σ2
xx+ (σxy− 1.5e2/h)2 = (0.5e2/h)2,
the transition point, denoted by Q in the figure, is not located at the top of the semicircle. Although the flow lines at both sides of Q do flow to opposite directions, starting from point
A, at which σxy > 1.5e2/h, the flow line moves toward to the point (e2/h, 0) rather than
(2e2/h, 0) with decreasing T , namely, the temperature-driven flow lines are distorted. Clearly,
the 2–1 transition we observed has the properties of the(2) group: valid semicircle relation, system-dependent transition point, and distorted flow diagram.
Figure 4. Curves ofσx xandσx ywith respect to the scaling parameter(ν − ν2−1)/Tκ, where the critical exponentκ = 0.4, ν is the filling factor, and ν2−1is the filling factor at the critical point.
At low T , magnetic-field-induced phase transitions are expected to follow the two-parameter scaling such that both σxx andσxy (or ρxx and ρxy) are functions of the scaling
parameter [5,6,9,10]. Figure4 shows the curves ofσxx andσxy at different temperatures
with respect to the scaling parameter s ≡ (ν − ν2−1)/Tκ withκ = 0.4, where ν2−1 is the
filling factor at B2−1. We can see in this figure that theσxycurves at different T collapse into a
single curve and henceσxyis a function of the scaling parameter in the whole transition region.
However, we found that although theσxx curves collapse nicely forν > ν2−1, they do not
merge for(ν − ν2−1)/Tκ < −0.017. When the scaling on σxx fails, we find that the T -driven
flow lines move away from the semicircle with increasing T . For example, the flow line reaches point C at T = 0.94 K in figure3. It is shown that in the effective Lagrangian that determines the properties at the critical points,σxy andσxx are coefficients of the topological and kinetic
terms, respectively [20,28]. The topological term is an important quantity for constructing the phase diagram of the quantum Hall effect [28,29], and it has been shown that the critical point inσxy is more robust than that inσxx with respect to the temperature T [30]. Our study shows
that the universal scaling onσxycan remain correct when the T -driven flow lines do not follow
the semicircle law and the scaling onσxx is invalid, and supports the claim that the critical point
inσxy is more robust than that inσxx with respect to the temperature T [30].
With increasing B, as reported in [18], we observed the 1–0 transition which follows the universal scaling and semicircle law, while the resistivity at the critical field B = 14.7 T is not universal. The critical magnetic field is close to the maximum field 15 T of our system, so we cannot investigate the details of the insulating phase such as whether it is terminated by a fractional quantum Hall liquid at higher B or not [31,32]. But the scaling behaviours near
B= 14.7 T show that we do observe the 1–0 transition. The T -driven flow lines near its critical
magnetic field are shown in figure5, in which the flow lines are of opposite directions near the point Q. Such a point corresponds to the critical point in the 1–0 transition and serves the same role as the point Q in the 2–1 transition. As shown in the inset of figure5, theσxx(σxy)
curves at T = 0.31–0.94 K are along the expected semicircle because of the semicircle law. The point Qdeviates from the top position of the semicircle, and the flow diagram also shows the features of(2) symmetry. The deviation on such a point, however, might either be due to the inappropriate conversion form resistivities to conductivities or the reduction of modular symmetry [9]. In the 1–0 transition reported in [33], the features of (2) symmetry are
J. Phys.: Condens. Matter 19 (2007) 026205 C F Huang et al
Figure 5. The temperature-driven flow diagram for the 1–0 transition near the critical point. The
dash–dot–dot line is the semicircle satisfyingσ2
x x+ (σx y− 0.5e2/h)2= (0.5e2/h)2and each solid
line corresponds to a temperature-driven flow line at a magnetic field. The symbols, ♦, , and
◦
are for the points at T= 0.94, 0.68, 0.49, and 0.31 K, respectively. The point Qcorresponds to the critical point of the 1–0 transition. In the insetσx x(σx y) at T = 0.94, 0.68, 0.49, and 0.31 K areplotted respectively, and the dash–dot–dot line is the semicircle.
removed by renormalizingρxx [9,34]. To further investigate the 1–0 transition, it is noted
that the particle–hole transformation is generalized as in equation (1) if the modular symmetry is reduced to(2). Under the generalized particle–hole transformation, the flow diagrams for the 2–1 and 1–0 transitions are symmetric with respect to the vertical lineσxy = e2/h [9].
We can see in figures5and3thatσxy < 0.5e2/h at Q, whileσxy > 1.5e2/h at Q, and the
values ofσxx are close at these two points. Therefore, the flow diagrams have the feature of
generalized particle–hole transformation, which indicates that the 1–0 transition follows(2) symmetry.
Using the dissipation equation [35–38], we estimated that the spin splitting in our sample is about 0.04 of the cyclotron energy from the value ofρxx at T = 0.94 K and 0.68 K at B = 9.8 T, where ρxx(B) reaches its minimum in the ν = 1 QH state. Such a value is
larger than the ratio of the bare spin gap to cyclotron splitting in the 2D GaAs system, which indicates the importance of the exchange enhancement on Zeeman spin splitting [25–27]. In [1], Shahar et al also reported a study on the 2–1 and 1–0 transitions in the 2DES in an AlGaAs/GaAs heterostructure. In their report, the carrier concentration and mobility of the sample are n = 2.27 × 1011 cm−2andμ = 1.08 × 104cm2V−1s−1. Although both n and
μ of their sample are similar to those in our sample, the modular symmetry is not reduced
from0(2) to (2) in their results. It should be noted that the effective spin splitting for QH
systems may be sample dependent [39]. Hence different modular symmetries might show up for samples with different effective spin gaps. In addition, it is in debate whether the dissipation equation can yield the spin gap [40–42]. More studies are necessary to see how to obtain the effective gap to determine the modular symmetry quantitatively.
To study the reduction of modular symmetry due to the small spin gap, the spin splitting has to be resolved enough to separate the Landau bands. On the other hand, the splitting also has to be so small that there exists coupling between two Landau bands separated by a spin gap. If the splitting is so large that the coupling can be ignored, we should observe spin-resolved transitions governed by0(2) symmetry just as in the report of Shahar et al [1]. But if
the spin splitting is too small and is unresolved, all (spin-degenerate) Landau bands are evenly separated and there should be no reduction of modular symmetry. Therefore, it is not easy to
Figure 6. The curves ofρx yin the 4–2 transition. The dash–dot–dot line labels the critical point
magnetic field B4−2= 2.46 T.
observe(2) symmetry. At low fields, as mentioned above, there is no QH state of the odd filling factor in our study and thus the spin splitting is unresolved. In figure1we can observe the 4–2 transition, which is a spin-degenerate transition. At the critical field B4−2of such a
transition,ρxx andρxy should equal 0.1h/e2and 0.3h/e2if the semicircle law is valid and the
unstable point in theσxy–σxx plane is at the top of the semicircle. Although in theν = 4 QH
stateρxxdoes not approach zero in our study, the quantum interference leading to QH states can
still induce plateaus inρxy [43]. In our study, we can identify B4−2 = 2.46 T at which ρxy is T -independent while inρxx there is no well-defined T -independent point. In figure6, at B4−2
the Hall resistivityρxx = 7.75 ± 0.05 k is close to the expected value of 0.3h/e2. Therefore,
the universality of critical Hall resistivity is valid in such a spin-degenerate transition, and the symmetry is the0(2) symmetry.
To further investigate the modular symmetry in the gated 2DES, we changed the gate voltage Vg from −0.1 to 0 V. A well-developed spin-resolved P–P transition is observed
between QH states ofν = 2 and 1, and the reduction of the modular symmetry can still be identified in such a transition by investigating the T -driven flow lines and critical points. In addition, the universal scaling is also observed inσxy even when the semicircle law and the
scaling onσxx both become invalid. On the other hand, the critical value ofρxy in the 4–2
transition remains at 0.3h/e2 when V
g is changed, which indicates there is no reduction of
modular symmetry in such a spin-degenerate transition. Thus we observe the reduction of modular symmetry from0(2) to (2) again at Vg= 0 V.
In conclusion, we have performed a magnetotransport study on a gated 2DES in an AlGaAs/GaAs heterostructure to study the universality of magnetic-field-induced phase transitions. A temperature-driven flow diagram is constructed by studying the 2–1 transition, and our study supports Dolan’s suggestion [9] that we should consider(2) symmetry under a small but resolved spin splitting, which breaks the0(2) symmetry. The reduction of the
symmetry from0(2) to (2) is due to the unequal energy splittings arising from the Zeeman
effect and Landau quantization. We also showed that the energy splitting is important to the scaling onσxx and the semicircle law over the same temperature range. The modular symmetry
can also be reduced in the 1–0 transition since its critical point can be related to that of the 2–1 transition by the generalized particle–hole transformation. The universal scaling onσxy, in
fact, can survive when the scaling onσxx is invalid and the temperature-driven flow lines are
J. Phys.: Condens. Matter 19 (2007) 026205 C F Huang et al
Acknowledgments
This work is supported by the National Science Council and Ministry of Education of the Republic of China. D R Hang acknowledges support from Aim for the Top University Plan, ACORC, NSYSU. One of the authors (CFH) thanks B P Dolan and C P Burgess for valuable discussions.
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