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Critical exponent in the fractional quantum Hall effect
Surface Science 229 North-Holland
13
( 1990) 13-l 5
CRITICAL
EXPONENT
IN THE FRACTIONAL
QUANTUM
L. ENGEL,
H.P. WEI, D.C. TSUI and M. SHAYEGAN
HALL EFFECT
Department ofElectrical Engineering, Princeton University, Princeton, NJ 08544, USA Received
18 July 1989; accepted
for publication
14 September
We report the observation of a universal critical exponent ature ( T) dependent electron transport in between Landau AlGaAs-GaAs heterostructure. Specifically, the maximum on T, i.e. 5 T --Ic.The critical exponent K= 0.43 k 0.02. This Hall effect (IQHE) [ Wei et al., Phys. Rev. Lett. 6 1 ( 1988) correlation length for the wavefunction of the quasi-particles be characterized by a universal critical exponent, K, in both
1989
in the fractional quantum Hall effect (FQHE) from a study of the temperlevel tilling factors v = $ and v = f , over a T range from I .2 K to 22 mK in an value of dp,,/dB and the width of pxr show the same power law dependence power law dependence is the same as that obtained in the integral quantum 12941, in analogy to which our result indicates the existence of a divergent in the FQHE for f < v ~3. The localization to delocalization transition can the IQHE and the FQHE.
The fractional quantum Hall effect (FQHE) [ 1] is manifestation of new many-body ground-states of two-dimensional electrons in strong magnetic fields (B), each of which is a uniform density, incompressible fluid, separated by an energy gap from its quasi-particle, quasi-hole excitations [ 2 1. The state at a rational fraction of a Landau level filling, u=f; is characterized by a quantized plateau in the Hall resistance (p,,,) about v and, concomitantly, a vanishing diagonal resistivity (p,) as T-0. Since the ground-state is an incompressible electron liquid, it can flow with no dissipation and the dissipative pxx arises from transport of the quasi-particles and quasiholes. The fractionalization of the charge carried by these quasi-particles and quasi-holes is the origin of the fractional quantization of the Hall conductance to a,,=jii*/h and their localization gives rise to the finite width of the quantized Hall plateau. Thus, the FQHE is phenomenologically very similar to the IQHE. The transport coefficients in between two adjacent Hall plateaus show similar T dependences. That is, when T decreases, the sharpness of the transition in pxY from one plateau to another increases and the width for finite non-zero pxx decreases. Recently, Wei et al. [ 31 have reported a detailed study of the T dependent transport in the IQHE regime. They have shown that a critical exponent can uniquely characterize the increasing sharpness with 0039-6028/90/$ (North-Holland)
03.50 0 Elsevier Science Publishers
B.V.
decreasing temperature of the transitions between two adjacent plateaus of the IQHE. Specifically, it is observed that there is a characteristic temperature, T,,, below which both the maximum value of dp,,/ dB [denoted by (dp,/dB)“““] and the reciprocal of the field-separation between the two extrema of dpxx/ dB [denoted by A-‘] diverge like T-” with decreasing T, where K= 0.42 k 0.04 independent of Landau levels. This result illustrates the theoretical prediction of scaling in the IQHE and that the single particle electron localization length diverges according to a universal power law, as the Fermi energy approaches the center of a Landau level [ 41. However, it is not clear whether the notion of scaling and this theoretical prediction of a divergent localization length can be extended to the FQHE. In this paper, we report a new experimental result on the FQHE, concerning the electron transport in between two adjacent fractional plateaus. Specifically, we perform the same study as in ref. [ 3 ] on the FQHE in AlGaAs-GaAs heterostructures. We find that between Landau level filling factors v = 3 and f , and A-’ diverge like T-” for both (dp,/dB)““” 22 mJ < T< 1.2 K. The critical component ~=0.43$_0.02 is the same as that for the IQHE reported in ref. [ 31. The data presented in this report are from a sample of AlGaAs-GaAs heterostructure that has mo-
L. Engel et al./Critical exponent in the fractional quantum Hall effect
r
I! 1i J
0
2
4
6
8
10
12
W7
0
2
4
6
8
10
12
WI
Fig. I. Transport coeffkients p,, and pu versus B at three different temperatures.
bility 1.2~ lo6 cmZ/V.s at 1.2 K and two-dimensional electron density 9.4 x 10” cme2. The transport coefficients pXXand pxy were measured as functions of the magnetic field at fixed temperatures between 22 mK and 1.2 K, using the AC lock-in technique. The current was typically 1 nA with frequency of around 15 Hz. The signals proportional to p,,(B), p,,,(B) were digitized, and a computer calculated the field-derivative quantities dp,/dB and dp,ldB to obtain (dp,,/dB)““” and A- ’ respectively. We plot the transport coefficients, pXv and pXX as a function of B for the sample at three different T’s in fig. 1. The plateaus at v= f and 3 are both well-developed over most of the experimental temperature range. The 3 minimum is visible even at 1.24 K, and remains resolved down to the lowest T. At no temperature there is any sign of developing higher order fractional states in between u= f and $ The Tdependent quantities (dp,,/dB)““” and A-’ forj
on T, i.e. - T-“. The best-fit rc values from d-’ (T) and (dp,,/dB)“““(T) are 0.421 kO.015 and 0.443 ?O.OlO respectively. These two values for K agree with one another to within about five percent. Within the experimental error, this measured critical exponent for the FQHE is the same as that for the IQHE reported in ref. [ 31. We have studied a total of three different samples, including one with lower
Fig. 2. Tdependence of (dpX~/dB)“““and A-’ for 4 -CUC;
L. Engel et al/Critical
exponent in the fractional
mobility ( = 400 000 cm’/V.s), and our result for K is lc=O.43 * 0.02. Despite the theoretical conjecture of scaling and some recent experimental investigation of it in the FQHE [ 5,6], nothing is known theoretically about the electron transport in between two adjacent fractional Hall plateaus. In the picture of Laughlin [ 21 and Halperin [ 71, the $ Hall plateau above v=i and the i Hall plateau below v= f are direct consequences of localization of the quasi-particles of e/5 charge and the quasi-holes of - e/ 3 charge, respectively. The nature of the extended states giving rise to dissipative pXXand the transition of pXYfrom the + plateau to the f plateau is not known. Our result explicitly demonstrates that, in analogy to the case in the IQHE, the correlation length for the wavefunction of extended states in between these two adjacent fractional plateaus diverges as T+O. This divergence can be characterized uniquely by a critical exponent IC.Furthermore, the critical exponent is the same as that in the IQHE, indicating that the localization to delocalization transitions in both cases belong to one universality class. In summary, we have measured T dependent divergent quantities (dp,,/dB)““” and A-’ for f
quantum Hall effect
15
of them diverge like T e-Kfor 22 mK < T-c 1.2 K. The critical exponent ~=0.43 k 0.02 is identical to that in the IQHE. To further test this universality, studies are now in progress to examine the transition from the 3 to 2 plateaus, and the transitions between integral and fractional quantum Hall plateaus. This work is supported in part by the NSF, grants DMR-8719694 and DMR-8705002, and by the ONR, grant NO00 14-89-J- 1567.
References 111 D.C. Tsui, H.C. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982)
1559.
[2 R.B. Laughlin, Phys. Rev. Lett. 50 (1983)
1395. H.P. Wei, DC. Tsui, M. Paalanen and A.M.M. Pruisken, Phys. Rev. Lett. 6 I ( 1988) 1294. [4 1 A.M.M. Pruisken, Phys. Rev. Lett. 61 (1988) 1297. 15 R.B. Laughlin, M.L. Cohen, J.M. Kosterlitz, H.M. Levine, S.B. Libby and A.M.M. Pruisken, Phys. Rev. B 32 (1985) 1311. 16 1 R.G. Clark, J.R. Mallett, S.R. Haynes, P.A. Maksym, J.J. Harris and C.T. Foxon, in: Proc. 19th Int. Conf. on the Physics of Semiconductors, Warsaw 1988, Ed. W. Zawadzki (IFPAS, Warsaw, 1989) p. 101. [7 B.I. Halperin, Helv. Phys. Acta 56 (1983) 75. [3
13
( 1990) 13-l 5
CRITICAL
EXPONENT
IN THE FRACTIONAL
QUANTUM
L. ENGEL,
H.P. WEI, D.C. TSUI and M. SHAYEGAN
HALL EFFECT
Department ofElectrical Engineering, Princeton University, Princeton, NJ 08544, USA Received
18 July 1989; accepted
for publication
14 September
We report the observation of a universal critical exponent ature ( T) dependent electron transport in between Landau AlGaAs-GaAs heterostructure. Specifically, the maximum on T, i.e. 5 T --Ic.The critical exponent K= 0.43 k 0.02. This Hall effect (IQHE) [ Wei et al., Phys. Rev. Lett. 6 1 ( 1988) correlation length for the wavefunction of the quasi-particles be characterized by a universal critical exponent, K, in both
1989
in the fractional quantum Hall effect (FQHE) from a study of the temperlevel tilling factors v = $ and v = f , over a T range from I .2 K to 22 mK in an value of dp,,/dB and the width of pxr show the same power law dependence power law dependence is the same as that obtained in the integral quantum 12941, in analogy to which our result indicates the existence of a divergent in the FQHE for f < v ~3. The localization to delocalization transition can the IQHE and the FQHE.
The fractional quantum Hall effect (FQHE) [ 1] is manifestation of new many-body ground-states of two-dimensional electrons in strong magnetic fields (B), each of which is a uniform density, incompressible fluid, separated by an energy gap from its quasi-particle, quasi-hole excitations [ 2 1. The state at a rational fraction of a Landau level filling, u=f; is characterized by a quantized plateau in the Hall resistance (p,,,) about v and, concomitantly, a vanishing diagonal resistivity (p,) as T-0. Since the ground-state is an incompressible electron liquid, it can flow with no dissipation and the dissipative pxx arises from transport of the quasi-particles and quasiholes. The fractionalization of the charge carried by these quasi-particles and quasi-holes is the origin of the fractional quantization of the Hall conductance to a,,=jii*/h and their localization gives rise to the finite width of the quantized Hall plateau. Thus, the FQHE is phenomenologically very similar to the IQHE. The transport coefficients in between two adjacent Hall plateaus show similar T dependences. That is, when T decreases, the sharpness of the transition in pxY from one plateau to another increases and the width for finite non-zero pxx decreases. Recently, Wei et al. [ 31 have reported a detailed study of the T dependent transport in the IQHE regime. They have shown that a critical exponent can uniquely characterize the increasing sharpness with 0039-6028/90/$ (North-Holland)
03.50 0 Elsevier Science Publishers
B.V.
decreasing temperature of the transitions between two adjacent plateaus of the IQHE. Specifically, it is observed that there is a characteristic temperature, T,,, below which both the maximum value of dp,,/ dB [denoted by (dp,/dB)“““] and the reciprocal of the field-separation between the two extrema of dpxx/ dB [denoted by A-‘] diverge like T-” with decreasing T, where K= 0.42 k 0.04 independent of Landau levels. This result illustrates the theoretical prediction of scaling in the IQHE and that the single particle electron localization length diverges according to a universal power law, as the Fermi energy approaches the center of a Landau level [ 41. However, it is not clear whether the notion of scaling and this theoretical prediction of a divergent localization length can be extended to the FQHE. In this paper, we report a new experimental result on the FQHE, concerning the electron transport in between two adjacent fractional plateaus. Specifically, we perform the same study as in ref. [ 3 ] on the FQHE in AlGaAs-GaAs heterostructures. We find that between Landau level filling factors v = 3 and f , and A-’ diverge like T-” for both (dp,/dB)““” 22 mJ < T< 1.2 K. The critical component ~=0.43$_0.02 is the same as that for the IQHE reported in ref. [ 31. The data presented in this report are from a sample of AlGaAs-GaAs heterostructure that has mo-
L. Engel et al./Critical exponent in the fractional quantum Hall effect
r
I! 1i J
0
2
4
6
8
10
12
W7
0
2
4
6
8
10
12
WI
Fig. I. Transport coeffkients p,, and pu versus B at three different temperatures.
bility 1.2~ lo6 cmZ/V.s at 1.2 K and two-dimensional electron density 9.4 x 10” cme2. The transport coefficients pXXand pxy were measured as functions of the magnetic field at fixed temperatures between 22 mK and 1.2 K, using the AC lock-in technique. The current was typically 1 nA with frequency of around 15 Hz. The signals proportional to p,,(B), p,,,(B) were digitized, and a computer calculated the field-derivative quantities dp,/dB and dp,ldB to obtain (dp,,/dB)““” and A- ’ respectively. We plot the transport coefficients, pXv and pXX as a function of B for the sample at three different T’s in fig. 1. The plateaus at v= f and 3 are both well-developed over most of the experimental temperature range. The 3 minimum is visible even at 1.24 K, and remains resolved down to the lowest T. At no temperature there is any sign of developing higher order fractional states in between u= f and $ The Tdependent quantities (dp,,/dB)““” and A-’ forj
on T, i.e. - T-“. The best-fit rc values from d-’ (T) and (dp,,/dB)“““(T) are 0.421 kO.015 and 0.443 ?O.OlO respectively. These two values for K agree with one another to within about five percent. Within the experimental error, this measured critical exponent for the FQHE is the same as that for the IQHE reported in ref. [ 31. We have studied a total of three different samples, including one with lower
Fig. 2. Tdependence of (dpX~/dB)“““and A-’ for 4 -CUC;
L. Engel et al/Critical
exponent in the fractional
mobility ( = 400 000 cm’/V.s), and our result for K is lc=O.43 * 0.02. Despite the theoretical conjecture of scaling and some recent experimental investigation of it in the FQHE [ 5,6], nothing is known theoretically about the electron transport in between two adjacent fractional Hall plateaus. In the picture of Laughlin [ 21 and Halperin [ 71, the $ Hall plateau above v=i and the i Hall plateau below v= f are direct consequences of localization of the quasi-particles of e/5 charge and the quasi-holes of - e/ 3 charge, respectively. The nature of the extended states giving rise to dissipative pXXand the transition of pXYfrom the + plateau to the f plateau is not known. Our result explicitly demonstrates that, in analogy to the case in the IQHE, the correlation length for the wavefunction of extended states in between these two adjacent fractional plateaus diverges as T+O. This divergence can be characterized uniquely by a critical exponent IC.Furthermore, the critical exponent is the same as that in the IQHE, indicating that the localization to delocalization transitions in both cases belong to one universality class. In summary, we have measured T dependent divergent quantities (dp,,/dB)““” and A-’ for f
quantum Hall effect
15
of them diverge like T e-Kfor 22 mK < T-c 1.2 K. The critical exponent ~=0.43 k 0.02 is identical to that in the IQHE. To further test this universality, studies are now in progress to examine the transition from the 3 to 2 plateaus, and the transitions between integral and fractional quantum Hall plateaus. This work is supported in part by the NSF, grants DMR-8719694 and DMR-8705002, and by the ONR, grant NO00 14-89-J- 1567.
References 111 D.C. Tsui, H.C. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982)
1559.
[2 R.B. Laughlin, Phys. Rev. Lett. 50 (1983)
1395. H.P. Wei, DC. Tsui, M. Paalanen and A.M.M. Pruisken, Phys. Rev. Lett. 6 I ( 1988) 1294. [4 1 A.M.M. Pruisken, Phys. Rev. Lett. 61 (1988) 1297. 15 R.B. Laughlin, M.L. Cohen, J.M. Kosterlitz, H.M. Levine, S.B. Libby and A.M.M. Pruisken, Phys. Rev. B 32 (1985) 1311. 16 1 R.G. Clark, J.R. Mallett, S.R. Haynes, P.A. Maksym, J.J. Harris and C.T. Foxon, in: Proc. 19th Int. Conf. on the Physics of Semiconductors, Warsaw 1988, Ed. W. Zawadzki (IFPAS, Warsaw, 1989) p. 101. [7 B.I. Halperin, Helv. Phys. Acta 56 (1983) 75. [3