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Strongly anisotropic electronic transport in higher Landau levels
Physica E 6 (2000) 36–39
www.elsevier.nl/locate/physe
Strongly anisotropic electronic transport in higher Landau levels R.R. Dua; b;∗ , W. Pana; c , H.L. Stormerd; e , D.C. Tsuia , L.N. Pfei erd , K.W. Baldwind , K.W. Westd a Princeton University, Princeton, NJ, USA of Physics, University of Utah, Salt Lake City, UT, USA c NHMFL, Tallahassee, FL, USA d Bell Labs, Lucent Technologies, Murray Hill, NJ, USA e Columbia University, New York, NY, USA
b Department
Abstract Low-temperature, electronic transport in higher Landau levels (N ¿ 1) in a two-dimensional electron system is strongly anisotropic. At half- lling of either spin level of such Landau levels ( = 92 ; 11 ; 13 ; 15 ; etc.) the magnetoresistance either 2 2 2 collapses to form a deep minimum or is peaked in a sharp maximum, depending on the in-plane current direction. The anisotropic axis can be reoriented by applying an in-plane magnetic eld of ∼1–2 T strength. The magnetoresistance at = 52 and = 72 (N = 1) is initially isotropic but an in-plane eld induces a strong anisotropy. Our observations are strong evidence for a new many-electron phase in higher Landau levels, which forms spontaneously or can be induced by an in-plane eld. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Magneto transport; Landau levels
1. Introduction Two-dimensional electron systems (2DES) in high magnetic eld exhibit a multitude of novel electronic phases [1]. The electronic states of the fractional quantum Hall e ect (FQHE) are the most abundant and probably also the best known. Most of the transport studies have been performed on the lowest Landau level, N = 0. Magnetotransport on higher Landau levels had been limited to the still enigmatic ∗ Corresponding author. Correspondence address: Department of Physics, University of Utah, Salt Lake City, UT, USA. Tel.: +1-801-581-6806; fax: +1-801-581-4801. E-mail address: [email protected] (R.R. Du)
even-denominator FQHE state at lling factor = 52 and features in its immediate vicinity in the N = 1 Landau level [2,3]. Only recently have experiments on N ¿ 1 Landau levels been reported [4 –7], indicating novel electronic phases. We report magnetotransport measurements [5,6] in higher Landau levels ( lling factor ¿ 2, i.e., N ¿ 1) from high-mobility 2DES in modulation-doped GaAs=AlGaAs heterostructures. Electronic transport in higher Landau levels, ¿ 4, di ers remarkably from the usual electronic transport in Landau levels 64. Low-temperature (T ¡ 100 mK) electronic transport at half- lling of either spin level ¿ 4 13 15 ( = 92 ; 11 2 ; 2 ; 2 , etc.) is strongly anisotropy. At these
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 9 3 - 4
R.R. Du et al. = Physica E 6 (2000) 36–39
half- llings the magnetoresistance either collapses to form a deep minimum or is peaked in a sharp maximum, depending on the in-plane current direction. Such anisotropies are absent in the ¡ 4 Landau level, which are dominated by the states of the FQHE. In the vicinity of quarter- llings in the ¿ 4 levels satellite minima (not unlike those from the FQHE), which do not show such anisotropy, are observed. However, these satellite minima forgo FQHE plateau formation in the Hall resistance, approaching instead integer values for their concomitant plateaus. The transport anisotropies may be indicative of a new many-electron state, which forms exclusively in higher Landau levels [8–12]. While the origin of these states remains unclear, they are believed to arise from the formation of a striped electronic phase or an electronic phase akin to a liquid crystal phase. The satellite minima and their associated integer Hall plateaus may be manifestations of novel phase not unlike a charge density wave. We have further investigated the in uence of an in-plane magnetic eld on these states under tilted magnetic eld. In the electrically anisotropic phase at = 92 and 11 2 an in-plane magnetic eld of ∼1–2 T overcomes its initial pinning to the crystal lattice and reorients this phase. In the initially isotropic phase at = 52 and 72 an in-plane magnetic eld induces a strong anisotropy. In all cases, for high in-plane elds, the high resistance axis is parallel to the direction of the in-plane eld. 2. Experiment The magnetotransport coecients were measured in a square specimen cleaved along the [1 1 0] and [1 1 0] directions from a MBE wafer. 1 Eight indium contacts were di used symmetrically around the edges of the sample, permitting the measurements of the Rxx and the Ryy from the same specimen. The only di erence between the two cases (Rxx and Ryy ) is the con guration of the current and voltage probes, which have ◦ been rotated by 90 (see schematics in Fig. 1). Exper1 The resistance anisotropy measured on a square sample is believed to be enhanced due to simple geometric e ects, S.H. Simon, cond-matt=9903086; this consideration would not impact our conclusion in this paper.
37
Fig. 1. Magnetoresistance along two perpendicular in-plane directions of a two-dimensional electron system at 60 mK. Inset: (left) temperature dependence of the 92 resistance maximum and 92 resistance minimum at = 92 versus inverse temperature; (right) temperature dependence of the magnetoresistance minimum around ∼ 4 + 14 for both Rxx and Ryy .
iments were performed in a dilution refrigerator immersed into a superconducting solenoid. In tilted eld experiments the sample is placed on a precision rotator inside the mixing chamber of the dilution refrigerator. The sample can be rotated in situ around an axis perpendicular to the eld. An in-plane magnetic eld, Bip = B × sin , is introduced when the magnetic eld, B, is tilted away (tilt angle ) from the sample normal. As usual, the perpendicular eld is determined by Bperp = B × cos . 3. Results and discussion ◦
The data for = 0 were measured from a sample having an electron density of n = 2:3 × 1011 =cm2 and a mobility of = 1:2 × 107 cm2 =V s. Fig. 1 shows Rxx and Ryy traces recorded at T = 60 mK between Landau level lling factor = 2 and = 12, i.e., for N = 1– 6. In the N = 1 Landau level (between = 2 and 4) the previously observed characteristic features of the FQHE are well developed. In particular, sharp
38
R.R. Du et al. = Physica E 6 (2000) 36–39
minima of the even-denominator FQHE states at = 52 and = 72 are clearly visible, in both traces. The most striking discrepancy between the N ¿ 1 and N = 1 data lies in their extraordinary dependence on the in-plane current direction. While the N = 1 FQHE features are essentially independent of the in-plane current direction, the minima at half- lling in N ¿ 1 Landau levels turn into a set of sharp peaks ◦ when the current is being rotated by 90 . In particular 9 the spikes at = 2 exceed all other features of the spectrum by almost a factor of two. The data in Rxx seem to be a superposition of Ryy and the sharp spikes at half- llings. We have observed the anisotropy seen for N ¿ 1 in several other samples of electron densities 1.1–2:9 × 1011 cm2 and mobilities 6.8–12:8 × 106 cm2 =V s. Beyond the strong minima at half- lling, additional satellite minima are visible in the N = 2 Landau level. Their positions do not seem to follow the sequence observed in the N = 1 level. It is dicult to associate these structures with a particular rational lling factor, but they are located in the vicinity of 14 and 34 lling. Comparison between the Rxx and the Ryy traces indicates that transport at these minima is isotropic. The temperatures dependence of the = 92 resistance is shown in the inset (left panel) in Fig. 1. Resistance is essentially isotropic above T = 100 mK but bifurcates into a rapid rise and rapid drop, depending on the in-plane current direction. In the range between T ∼ 100 and 60 mK, the behavior of both the Rxx and the Ryy can be described by an exponential R ˙ exp(±E=kT ), with an energy scale of E ∼ 0:55 K. The temperature dependence of the satellite minima is not unlike that of the FQHE. The right panel quanti es the temperature dependence of one of the satellite minima (close to 14 lling) in a standard Arrhenius plot for both Rxx and Ryy . Both show the same exponential drop in resistance with an activation energy of = 1:7 ± 0:2 K, where R ˙ exp(−=2kT ). However, its associated Hall resistance, RH , is unusual [5]. The concomitant RH value converges towards the nearest integer (i.e., i = 4), rather than the usual fractional, quantum Hall plateau. The tilted eld experiments were performed using a similar specimen, having a yet higher mobility of = 1:7 × 107 cm2 =V s. The sample was mounted in two
Fig. 2. Dependence of the magnetoresistance Rxx and Ryy around lling factor 92 and 11 as well as around 52 and 72 on angle, , in 2 a tilted eld where an in-plane eld is pointing the easy direction. Temperature T ∼ 40 mK.
di erent con gurations onto the rotator, either placing the Bip along the x-axis (“hard direction”), or along the y-axis (“easy direction”). As an example, Fig. 2 shows Rxx and Ryy data taken for Bip along the easy direction, y, of the anisotropic state. The insets depicts the geometry. With an increasing Bip , Rxx collapses and develops into a minimum, while Ryy rises and becomes a maximum at the ◦ highest tilt, = 74:3 . This can be viewed as if the ◦ anisotropic axis has been rotated by 90 . 5 7 For = 2 and 2 , in the absence of tilt, the data show practically no anisotropy. However, tilting of the sample and the associated increase of Bip drastically alters the data and introduces a strong anisotropy between Rxx and Ryy . As shown in Fig. 3, as Bip increases along the easy direction, Rxx decreases, while Ryy increases. In the situation where the Bip is along the hard direction, the in-plane eld preserves the 92 and 11 2 anisotropies while the 52 and 72 develop anisotropies in a similar fashion as the previous case. Fig. 3 summarizes the anisotropies for the strongest of states at = 92 and = 52 for both Bip directions. In all cases, for high in-plane elds, the high resistance axis is parallel to the direction of the in-plane eld. In summary, electronic transport in the higher Landau levels, N ¿ 1, di ers in several ways from the usual electronic transport in Landau levels N 61. Whereas in the lowest Landau levels the standard features of the FQHE dominate, transport around
R.R. Du et al. = Physica E 6 (2000) 36–39
39
is being viewed as consistent with a phase transition from a quantum Hall liquid to a strongly anisotropic phase [13,14]. Acknowledgements All transport experiments were performed at the Francis Bitter Magnet Lab in Cambridge, Massachusetts, and National High Magnetic Field Lab in Tallahassee, Florida. R.R.D., W.P. and D.C.T. are supported by AFOSR, NSF, and DOE. References Fig. 3. Amplitudes of Rxx and Ryy at = of in-plane magnetic eld Bip .
9 2
and =
5 2
as a function
half- lling in N ¿ 1 Landau levels is extremely anisotropic, developing large maxima or deep minima depending on the in-plane current direction. Satellite minima in the vicinity of quarter- lling do not show this anisotropy, but the concomitant Hall resistance converges to nearest integer Hall plateau. The nature of the state at = 52 is believed to be quite distinct from the state at = 92 . For one, the former is a true FQHE state [1–3] with plateau formation in RH , whereas such a plateau seems to be absent for the latter. And secondly, dramatic anisotropies in electronic transport in purely perpendicular magnetic eld have only been observed for the states at = 92 and equivalent, whereas they are absent in the = 52 state. It is remarkable that anisotropies not unlike those of the = 92 state can be induced in the = 52 state at suciently high in-plane magnetic eld. This result
[1] S. DasSarma, A. Pinczuk (Eds.), Perspectives in Quantum Hall E ects, Wiley, New York, 1997. [2] R.L. Willett, J.P. Eisenstein, H.L. Stormer, D.C. Tsui, A.C. Gossard, J.H. English, Phys. Rev. Lett. 59 (1987) 1776. [3] J.P. Eisenstein, R.L. Willett, H.L. Stormer, D.C. Tsui, A.C. Gossard, J.H. English, Phys. Rev. Lett. 66 (1988) 997. [4] M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 82 (1999) 394. [5] R.R. Du, D.C. Tsui, H.L. Stormer, L.N. Pfei er, K.W. Baldwin, K.W. West, Solid State Commun. 109 (1999) 389. [6] W. Pan, R.R. Du, H.L. Stormer, D.C. Tsui, L.N. Pfei er, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 83 (1999) 820. [7] M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 83 (1999) 824. [8] A.A. Koulakov, M.M. Fogler, B.I. Shklovskii, Phys. Rev. Lett. 76 (1996) 499. [9] R. Moessner, J.T. Chalker, Phys. Rev. B 54 (1996) 5006. [10] E. Fradkin, S.A. Kivelson, Phys. Rev. B 59 (1999) 8065. [11] H. Fertig, Phys. Rev. Lett. 82 (1999) 3693. [12] E.H. Rezayi, F.D.M. Haldane, K. Yang, Phys. Rev. Lett. 83 (1999) 1219. [13] T. Jungwirth, A.H. MacDonald, L. Smrcka, S.M. Girvin, preprint (cond-mat=9905353). [14] E.H. Rezayi, F.D.M. Haldane, preprint (cond-mat=9906137).
www.elsevier.nl/locate/physe
Strongly anisotropic electronic transport in higher Landau levels R.R. Dua; b;∗ , W. Pana; c , H.L. Stormerd; e , D.C. Tsuia , L.N. Pfei erd , K.W. Baldwind , K.W. Westd a Princeton University, Princeton, NJ, USA of Physics, University of Utah, Salt Lake City, UT, USA c NHMFL, Tallahassee, FL, USA d Bell Labs, Lucent Technologies, Murray Hill, NJ, USA e Columbia University, New York, NY, USA
b Department
Abstract Low-temperature, electronic transport in higher Landau levels (N ¿ 1) in a two-dimensional electron system is strongly anisotropic. At half- lling of either spin level of such Landau levels ( = 92 ; 11 ; 13 ; 15 ; etc.) the magnetoresistance either 2 2 2 collapses to form a deep minimum or is peaked in a sharp maximum, depending on the in-plane current direction. The anisotropic axis can be reoriented by applying an in-plane magnetic eld of ∼1–2 T strength. The magnetoresistance at = 52 and = 72 (N = 1) is initially isotropic but an in-plane eld induces a strong anisotropy. Our observations are strong evidence for a new many-electron phase in higher Landau levels, which forms spontaneously or can be induced by an in-plane eld. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Magneto transport; Landau levels
1. Introduction Two-dimensional electron systems (2DES) in high magnetic eld exhibit a multitude of novel electronic phases [1]. The electronic states of the fractional quantum Hall e ect (FQHE) are the most abundant and probably also the best known. Most of the transport studies have been performed on the lowest Landau level, N = 0. Magnetotransport on higher Landau levels had been limited to the still enigmatic ∗ Corresponding author. Correspondence address: Department of Physics, University of Utah, Salt Lake City, UT, USA. Tel.: +1-801-581-6806; fax: +1-801-581-4801. E-mail address: [email protected] (R.R. Du)
even-denominator FQHE state at lling factor = 52 and features in its immediate vicinity in the N = 1 Landau level [2,3]. Only recently have experiments on N ¿ 1 Landau levels been reported [4 –7], indicating novel electronic phases. We report magnetotransport measurements [5,6] in higher Landau levels ( lling factor ¿ 2, i.e., N ¿ 1) from high-mobility 2DES in modulation-doped GaAs=AlGaAs heterostructures. Electronic transport in higher Landau levels, ¿ 4, di ers remarkably from the usual electronic transport in Landau levels 64. Low-temperature (T ¡ 100 mK) electronic transport at half- lling of either spin level ¿ 4 13 15 ( = 92 ; 11 2 ; 2 ; 2 , etc.) is strongly anisotropy. At these
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 9 3 - 4
R.R. Du et al. = Physica E 6 (2000) 36–39
half- llings the magnetoresistance either collapses to form a deep minimum or is peaked in a sharp maximum, depending on the in-plane current direction. Such anisotropies are absent in the ¡ 4 Landau level, which are dominated by the states of the FQHE. In the vicinity of quarter- llings in the ¿ 4 levels satellite minima (not unlike those from the FQHE), which do not show such anisotropy, are observed. However, these satellite minima forgo FQHE plateau formation in the Hall resistance, approaching instead integer values for their concomitant plateaus. The transport anisotropies may be indicative of a new many-electron state, which forms exclusively in higher Landau levels [8–12]. While the origin of these states remains unclear, they are believed to arise from the formation of a striped electronic phase or an electronic phase akin to a liquid crystal phase. The satellite minima and their associated integer Hall plateaus may be manifestations of novel phase not unlike a charge density wave. We have further investigated the in uence of an in-plane magnetic eld on these states under tilted magnetic eld. In the electrically anisotropic phase at = 92 and 11 2 an in-plane magnetic eld of ∼1–2 T overcomes its initial pinning to the crystal lattice and reorients this phase. In the initially isotropic phase at = 52 and 72 an in-plane magnetic eld induces a strong anisotropy. In all cases, for high in-plane elds, the high resistance axis is parallel to the direction of the in-plane eld. 2. Experiment The magnetotransport coecients were measured in a square specimen cleaved along the [1 1 0] and [1 1 0] directions from a MBE wafer. 1 Eight indium contacts were di used symmetrically around the edges of the sample, permitting the measurements of the Rxx and the Ryy from the same specimen. The only di erence between the two cases (Rxx and Ryy ) is the con guration of the current and voltage probes, which have ◦ been rotated by 90 (see schematics in Fig. 1). Exper1 The resistance anisotropy measured on a square sample is believed to be enhanced due to simple geometric e ects, S.H. Simon, cond-matt=9903086; this consideration would not impact our conclusion in this paper.
37
Fig. 1. Magnetoresistance along two perpendicular in-plane directions of a two-dimensional electron system at 60 mK. Inset: (left) temperature dependence of the 92 resistance maximum and 92 resistance minimum at = 92 versus inverse temperature; (right) temperature dependence of the magnetoresistance minimum around ∼ 4 + 14 for both Rxx and Ryy .
iments were performed in a dilution refrigerator immersed into a superconducting solenoid. In tilted eld experiments the sample is placed on a precision rotator inside the mixing chamber of the dilution refrigerator. The sample can be rotated in situ around an axis perpendicular to the eld. An in-plane magnetic eld, Bip = B × sin , is introduced when the magnetic eld, B, is tilted away (tilt angle ) from the sample normal. As usual, the perpendicular eld is determined by Bperp = B × cos . 3. Results and discussion ◦
The data for = 0 were measured from a sample having an electron density of n = 2:3 × 1011 =cm2 and a mobility of = 1:2 × 107 cm2 =V s. Fig. 1 shows Rxx and Ryy traces recorded at T = 60 mK between Landau level lling factor = 2 and = 12, i.e., for N = 1– 6. In the N = 1 Landau level (between = 2 and 4) the previously observed characteristic features of the FQHE are well developed. In particular, sharp
38
R.R. Du et al. = Physica E 6 (2000) 36–39
minima of the even-denominator FQHE states at = 52 and = 72 are clearly visible, in both traces. The most striking discrepancy between the N ¿ 1 and N = 1 data lies in their extraordinary dependence on the in-plane current direction. While the N = 1 FQHE features are essentially independent of the in-plane current direction, the minima at half- lling in N ¿ 1 Landau levels turn into a set of sharp peaks ◦ when the current is being rotated by 90 . In particular 9 the spikes at = 2 exceed all other features of the spectrum by almost a factor of two. The data in Rxx seem to be a superposition of Ryy and the sharp spikes at half- llings. We have observed the anisotropy seen for N ¿ 1 in several other samples of electron densities 1.1–2:9 × 1011 cm2 and mobilities 6.8–12:8 × 106 cm2 =V s. Beyond the strong minima at half- lling, additional satellite minima are visible in the N = 2 Landau level. Their positions do not seem to follow the sequence observed in the N = 1 level. It is dicult to associate these structures with a particular rational lling factor, but they are located in the vicinity of 14 and 34 lling. Comparison between the Rxx and the Ryy traces indicates that transport at these minima is isotropic. The temperatures dependence of the = 92 resistance is shown in the inset (left panel) in Fig. 1. Resistance is essentially isotropic above T = 100 mK but bifurcates into a rapid rise and rapid drop, depending on the in-plane current direction. In the range between T ∼ 100 and 60 mK, the behavior of both the Rxx and the Ryy can be described by an exponential R ˙ exp(±E=kT ), with an energy scale of E ∼ 0:55 K. The temperature dependence of the satellite minima is not unlike that of the FQHE. The right panel quanti es the temperature dependence of one of the satellite minima (close to 14 lling) in a standard Arrhenius plot for both Rxx and Ryy . Both show the same exponential drop in resistance with an activation energy of = 1:7 ± 0:2 K, where R ˙ exp(−=2kT ). However, its associated Hall resistance, RH , is unusual [5]. The concomitant RH value converges towards the nearest integer (i.e., i = 4), rather than the usual fractional, quantum Hall plateau. The tilted eld experiments were performed using a similar specimen, having a yet higher mobility of = 1:7 × 107 cm2 =V s. The sample was mounted in two
Fig. 2. Dependence of the magnetoresistance Rxx and Ryy around lling factor 92 and 11 as well as around 52 and 72 on angle, , in 2 a tilted eld where an in-plane eld is pointing the easy direction. Temperature T ∼ 40 mK.
di erent con gurations onto the rotator, either placing the Bip along the x-axis (“hard direction”), or along the y-axis (“easy direction”). As an example, Fig. 2 shows Rxx and Ryy data taken for Bip along the easy direction, y, of the anisotropic state. The insets depicts the geometry. With an increasing Bip , Rxx collapses and develops into a minimum, while Ryy rises and becomes a maximum at the ◦ highest tilt, = 74:3 . This can be viewed as if the ◦ anisotropic axis has been rotated by 90 . 5 7 For = 2 and 2 , in the absence of tilt, the data show practically no anisotropy. However, tilting of the sample and the associated increase of Bip drastically alters the data and introduces a strong anisotropy between Rxx and Ryy . As shown in Fig. 3, as Bip increases along the easy direction, Rxx decreases, while Ryy increases. In the situation where the Bip is along the hard direction, the in-plane eld preserves the 92 and 11 2 anisotropies while the 52 and 72 develop anisotropies in a similar fashion as the previous case. Fig. 3 summarizes the anisotropies for the strongest of states at = 92 and = 52 for both Bip directions. In all cases, for high in-plane elds, the high resistance axis is parallel to the direction of the in-plane eld. In summary, electronic transport in the higher Landau levels, N ¿ 1, di ers in several ways from the usual electronic transport in Landau levels N 61. Whereas in the lowest Landau levels the standard features of the FQHE dominate, transport around
R.R. Du et al. = Physica E 6 (2000) 36–39
39
is being viewed as consistent with a phase transition from a quantum Hall liquid to a strongly anisotropic phase [13,14]. Acknowledgements All transport experiments were performed at the Francis Bitter Magnet Lab in Cambridge, Massachusetts, and National High Magnetic Field Lab in Tallahassee, Florida. R.R.D., W.P. and D.C.T. are supported by AFOSR, NSF, and DOE. References Fig. 3. Amplitudes of Rxx and Ryy at = of in-plane magnetic eld Bip .
9 2
and =
5 2
as a function
half- lling in N ¿ 1 Landau levels is extremely anisotropic, developing large maxima or deep minima depending on the in-plane current direction. Satellite minima in the vicinity of quarter- lling do not show this anisotropy, but the concomitant Hall resistance converges to nearest integer Hall plateau. The nature of the state at = 52 is believed to be quite distinct from the state at = 92 . For one, the former is a true FQHE state [1–3] with plateau formation in RH , whereas such a plateau seems to be absent for the latter. And secondly, dramatic anisotropies in electronic transport in purely perpendicular magnetic eld have only been observed for the states at = 92 and equivalent, whereas they are absent in the = 52 state. It is remarkable that anisotropies not unlike those of the = 92 state can be induced in the = 52 state at suciently high in-plane magnetic eld. This result
[1] S. DasSarma, A. Pinczuk (Eds.), Perspectives in Quantum Hall E ects, Wiley, New York, 1997. [2] R.L. Willett, J.P. Eisenstein, H.L. Stormer, D.C. Tsui, A.C. Gossard, J.H. English, Phys. Rev. Lett. 59 (1987) 1776. [3] J.P. Eisenstein, R.L. Willett, H.L. Stormer, D.C. Tsui, A.C. Gossard, J.H. English, Phys. Rev. Lett. 66 (1988) 997. [4] M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 82 (1999) 394. [5] R.R. Du, D.C. Tsui, H.L. Stormer, L.N. Pfei er, K.W. Baldwin, K.W. West, Solid State Commun. 109 (1999) 389. [6] W. Pan, R.R. Du, H.L. Stormer, D.C. Tsui, L.N. Pfei er, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 83 (1999) 820. [7] M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfei er, K.W. West, Phys. Rev. Lett. 83 (1999) 824. [8] A.A. Koulakov, M.M. Fogler, B.I. Shklovskii, Phys. Rev. Lett. 76 (1996) 499. [9] R. Moessner, J.T. Chalker, Phys. Rev. B 54 (1996) 5006. [10] E. Fradkin, S.A. Kivelson, Phys. Rev. B 59 (1999) 8065. [11] H. Fertig, Phys. Rev. Lett. 82 (1999) 3693. [12] E.H. Rezayi, F.D.M. Haldane, K. Yang, Phys. Rev. Lett. 83 (1999) 1219. [13] T. Jungwirth, A.H. MacDonald, L. Smrcka, S.M. Girvin, preprint (cond-mat=9905353). [14] E.H. Rezayi, F.D.M. Haldane, preprint (cond-mat=9906137).