Solid State Communications, Vol. 50, No. 9, pp. 841844, 1984. Printed in Great Britain.
00381098/84 $3.00 + .00 Pergamon Press Ltd.
DISORDER AND THE FRACTIONAL QUANTUM HALL EFFECT M.A. Paalanen, D.C. Tsui,*A.C. Gossard and J.C.M. Hwang t AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A. and *Department of Electrical Engineering and Computer Science, Princeton University, Princeton, NJ 08544, U.S.A.
(Received 24 February 1984 by A.G. Chynoweth) The fractional quantum Hall effect is studied in the 2D electron gas of four GaAsAlxGal_xAs heterostructures. Localization due to disorder, known to give rise to the wide integral quantum Hall plateaus, is demonstrated to inhibit the fractional effect, which is observed only in the higher mobility samples.
THE INTEGRAL QUANTUM Hall effect has been observed in several twodimensional (2D) electronic systems in strong perpendicular magnetic fields [ 13]. The exact quantization of the Hall resistance RH = Pxy and the disappearance of the diagonal resistivity Pxx over wide regions of field and density are explained by the localization of the single electron states in the Landau level tails [4, 5]. According to theory [5], the existence of exactly quantized Hall plateaus implies the existence of a finite number of extended states in every Landau level beneath E F to carry the Hall current. The fractional quantum Hall effect, on the other hand, has been found only in the extremely high mobility 2D electron gas in the GaAsAlxGa1_xAs heterostructures, which are known to have a high degree of lattice and interfacial perfection [6, 7]. It is associated with the formation of a new electronic state arising from the strong electronelectron interaction. A theory due to Laughlin [8] predicts the existence of fractionally charged excitations and a gap in the excitation spectrum, in qualitative agreement with experiments. As in the case of the integral quantum Hall effect, the disorder is thought to localize the fractionally charged excitations and to give rise to the finite width in the fractional Hall plateaus. Consequently, it reduces the observed activation energy from the theoretical prediction, which does not include disorder. This picture is consistent with existing data, which indicate that higher mobility samples have a larger 1/3 excitation energy [9]. More recently Chang et al. [10], were able to measure the excitation energy of the 2/3 state continuously as a function of the mobility and their data predict, at
t Present address: G. E. Electronics Laboratory, P.O. Box 488, Syracuse, NY 13221, U.S.A.
B ~ 6 T, a threshold mobility ta ~40 m 2 Vsec 1 , below which the 2/3 quantum state ceases to exist. In this paper, we report further magnetotransport studies of the lowest Landau level (i.e. the N = 0 and spin i" level) in four GaAsAlo.3Gao.TAs samples with ta = 2.4, 3.5, 7.1 and 4 9 m 2 Vsec 1 and n = 1.0, 1.2, 0.82 and 0.97 x 10 is m 2, respectively. For comparison, we also present data of the third lowest Landau level (1, t) in a fifth sample with ta = 8.6m 2 Vsec 1 and n = 4.0 x 10 is m 2. These results were obtained at low temperatures (65770 inK) and in magnetic fields up to B = 10.5 T. Based on the activated behavior O f p x y , we find the number of extended states of the (1, t) level to be small. This agrees with the previous observation of wide integral Hall plateaus [ 11 ]. The lowest attainable filling factor, defined by p = nh/eB, was less than 1/2 for all four low density samples and less than 1/3 for the lowest density sample. We observe the competition between the localization of states in the tails of the (0, t) level and the formation of the 1/3 and 2/3 fractional quantum states. In the lowest mobility sample (with ta = 2.4 m 2 Vsec 1 ), which shows no fractional effect, both Pxx and pxy of the (0, l') level show only activated behavior, similar to that observed in higher Landau levels. An estimate of the Landau level width is made from the low field Subnikovde Haas (SdH) oscillations and is compared with the predicted energy gap of the fractional quantum Hail state. Our samples were GaAsAlo.3Gao:As heterostructures, consisting of a 1 tam thick undoped GaAs layer, an undoped Alo.3Gao.TAs layer, a Si doped Alo.3Gao.TAs layer, and a thin cap layer, sequentially grown on Crdoped GaAs substrates by molecular beam epitaxy. The undoped A1GaAs layer, varying from 100 to 400 )k, separates the impurity scatterers from the 2D 841
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DISORDER AND THE FRACTIONAL QUANTUM HALL EFFECT
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inverse temperature T 1 for the high density GaAs sample at (1, ?) Landau level. electron gas in the interface between GaAs and A1GaAs. The electron density was determined from SdH oscillations of the 2D electrons, and from low field Hall measurements. Agreement between the two measurements indicates that no parallel conduction existed in our samples. Figure 1 illustrates the notations used to describe the lowest Landau level at high magnetic fields. The distance between neighboring levels with the same spin direction is h o ~ c and the spin splitting isg~BB. Here co c is the cyclotron frequency given by coe = eB/m*, g is the effective electron gfactor, and/JB iS the Bohr magneton. The broadening depends on the disorder and the magnetic field. The electron states in the tails of the levels are localized and the Hall current is carried by the extended states represented by the shaded regions. The existence o f exactly quantized Hall resistance plateaus implies the existence of a finite number
of extended states in every Landau level below the Fermi level E F [5]. In the single electron picture, the extended states and the localized states are separated at well defined energies Ee. In the extreme quantum limit, only the lowest Landau level is partially occupied. The fractional quantum Hall effect, observed in the high mobility samples, results from the formation of a new ground state due to the strong electronelectron interaction. In Laughlin's theory [8], the ground state is an incompressible quantum fluid with fractionally charged excitations separated in energy by a gap given by C2
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where lo = x/h/eB is the cyclotron radius. In Fig. 2 we show the temperature dependence of the Hall conductance oxy for the high density sample at filling of the (1,1") Landau level. At u = 2.57, which is slightly over half filling we find oxy temperature independent, indicating that E F is in the region of the extended states of the (1, f) level. At higher filling, as well as at lower filling, we find that (oxy  3e2/h) and (o,,y  2e2/h) are thermally activated, indicating localization of the electron states at the Fermi level. The activation energy increases with increasing [u  2.57[, in agreement with the picture that the activation energy
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DISORDER AND THE FRACTIONAL QUANTUM HALL EFFECT
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Fig. 5. The width of Landau levels obtained from the low field SdH oscillations at T = 0.06 K. The solid line is calculated from equation (2) at corresponding magnetic fields. The broken line is the width calculated from equation (2) at 10 T and Ao is the gap of the fractional quantum state also at 10T [equation (1)].
measures the distance between E F and the closest mobility edge E c (Fig. 1). We estimate from the data in Fig. 2 that less than 5% of the electron states in the (1, T) level are extended. Sample inhomogeneties make a more accurate estimation meaningless. Figure 3 shows the magnetic field dependence of px~ and Pxy, taken from three of our low density samples, demonstrating the effect of disorder on the fractional quantum Hall effect. The disorder in the 2D system is characterized by the sample mobility, which is, from top to bottom,/a = 2.4, 7.1 and 49 m 2 Vsec'. Several aspects of the data are apparent from Fig. 3. First the integer i = 1 quantized Hall plateau (i.e. Pxy = h/e2), corresponding to the complete filling of the lowest Landau level is clearly resolved in all our samples. The plateau is wider in the lower mobility samples, esoecially pronounced in those where fractionally quantized plateaus at u < 1 are not observed. This result is in contrast to that obtained from the 2D electrons in Siinversion layers, where neither the fractional quantum effect has been observed nor the i = I plateau is resolved. Second, the high mobility sample in the middle panel shows a dip in Px~ and a change in slope in Pxy at about u = ½. This behavior is characteristic of the ~ quantum state. Third, the ] state, which is not observed in the top two panels, is clearly resolved in the highest mobility sample in the bottom panel, where Pxx reaches a deep minimum and Oxy a plateau about u = 3. This result agrees with Chang et al., who
demonstrated the existence of a mobility threshold for the existence of the 2/3 quantum state. These qualitative aspects of the data demonstrate conclusively that increasing disorder inhibits the formation of the fractional quantum states. In Fig. 4, we further demonstrate the effect of disorder in the u < 1 limit, by showing the inverse T dependence of Pxx and Pxy shown in the top two panels of Fig. 3. For v > 0.5, Pxy as well as Pxx are seen to decrease with decreasing T in both samples. This thermally activated behavior results from the increase in the width of the i = 1 Hall plateau and its associated zero in Pxx The approximate temperature independence o f p x x and Pxy observed at u ~ 0.5, indicates that the i = 1 Hall plateau extends to p = 1/2 as T approaches zero. This result is consistent with our physical picture that the states at EF are extended when the lowest Landau level is approximately half filled and localized at larger filling. In both samples, the states are localized at u = 2/3 and the localization, resulting in a wide i = 1 plateau, inhibits the ~ fractional quantum effect. For v < 0.5, the two samples exhibit quite different characteristics. In the low mobility sample (upper panel), Pxx depends exponentially on I/T. The associated activation energy increases with decreasing u and at v = 0.37 it is ~ 0.6 K. Moreover, Pxy also shows thermally activated behavior, as expected from localization of the states at Ei~ and the transport is then being carried out by activation into the extended states. The
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DISORDER AND THE FRACTIONAL QUANTUM HALL EFFECT
fact that Oxy at u = 0.37 and ~ 0.3 K is much larger than the quantized Hall resistance for the ~ state (i.e. p:,y > 3hie 2) indicates that the ~ quantum case is also not observable in this low mobility sample. On the other hand, this characteristic T dependence is not observed in the data from the higher mobility sample shown in the lower panel. In this sample, the disorder is sufficiently strong to inhibit the ] state at B "~ 5 T, but not the ~ state at B ~ 10 T. Our data can qualitatively be understood by assuming that disorder reduces the energy gap, separating the ground state from its fractionally charged excitations, and that the fractional quantum states cease to exist when the gap vanishes. However, it is not clear how the disorder affects the interaction and how it reduces the energy gap. Chang et al., interpreted their data in terms of an excitation spectrum illustrated in the inset of Fig. 1. They assumed that disorder broadens the quasielectron and quasihole states into two bands of extended states separated by a gap filled with localized states. In this picture, increasing disorder will cause collapse of the gap by increasing broadening and/or by decreasing 2A0. We should note that, in the absence of a magnetic field, the mobility is a convenient parameter to characterize the disorder. In our high field limit, however, the mobility loses its physical meaning and the corresponding parameter to characterize the disorder is the width of the Landau levels. To estimate the width of the Landau levels from the amplitude of the SdH oscillations (Fig. 3), we set the full width 2P equal to hco e = heB/m* at the magnetic field B, where the peak to peak amplitude of the SdH oscillations is 50% of the zero field resistance. The resulting P is plotted in Fig. 5 for our four low density samples. In Ando's self consistent Born approximation, P, for short range scatterers is given by [12] P = h~c(2/rr/aB) 1/2,
(2)
independent of the Landau level number, but dependent on the magnetic field and mobility. The solid line in Fig. 5 is obtained from equation (2) using/.tB = c~cz0 "~ 1.8, where the calculated peak to peak amplitude of the SdH oscillations is 50% of the resistance at B = 0 [ 12]. The measured width deviates more severely from the solid line at the higher mobilities, consistent with the notion that inhomogeneous broadening by long range potential fluctuations is important in the high mobility
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samples. In the long range limit P is predicted to be independent of magnetic field and Landau level number [ 12]. For comparison, we also calculated V from equation (2) (valid for short range scatterers) for B = 10T, where the ~ state is expected to occur in our samples. This is shown as the broken curve, which is considerably above the experimental points. For our /J = 7.1 m 2 Vsec 1 sample, P ~ 19K, which is about 7Ao, and our low field SdH estimate gives Y "" 6.5 K, which is about 2Ao, where Ao is the energy gap of the state given by equation (1). In conclusion we have studied the effect of disorder on the recently discovered fractional quantum Hall states at v = ½ and 3 For our low density samples with n "" 10 Is m 2 we find that a sample mobility o f ~ 5 m 2 Vsec 1 is needed to observe the ~ state. At lower mobilities, both Pxx and p:,y are activated, signaling the localization of single electron states. We also estimate the width of the Landau levels and find inhomogeneous broadening due to long range potential fluctuations to be more important in high mobility samples. The Landau level broadening at which the ~ fractional quantum state is just resolved is about twice the energy gap predicted by Laughlin.
Acknowledgements  We thank Drs A.M. Chang and H.L. StOrmer for discussions. The work at Princeton University is supported by the Office of Naval Research and a grant from the National Bureau of Standards. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
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