The observation of new insulating phases of two-dimensional electrons in the
first excited Landau level is reported. These states, which are manifested as
re-entrant integer quantized Hall effects, exist alongside well-developed
even-denominator fractional quantized Hall states at nu=7/2 and 5/2 and new
odd-denominator states at nu=3+1/5 and 3+4/5.Comment: 4 pages, 3 figure

A. A. Koulakov

A. H. MacDonald

E. H. Rezayi

F. D. M. Haldane

H. C. Manoharan

H.-W. Jiang

J. P. Eisenstein

K. B. Cooper

K. W. West

L. N. Pfeiffer

M. M. Fogler

M. P. Lilly

N. d'Ambrumenil

N. Shibata

R. Du

R. L. Willett

R. Moessner

W. Pan

English

arXiv

The observation of new insulating phases of two-dimensional electrons in the first excited Landau level is reported. These states, which are manifested as reentrant integer quantized Hall effects, exist alongside well-developed even-denominator fractional quantized Hall states at n 7͞2 and 5͞2 and new odd-denominator states at n 3 1 1 5 and 3 1 4 5 . DOI: 10.1103/PhysRevLett.88.076801  PACS numbers: 73.43.Qt, 73.20.Qt, 73.63.Hs    At zero magnetic field Coulomb interactions in twodimensional electron systems (2DES) are less important than kinetic effects at all but the lowest densities. A large magnetic field B applied perpendicular to the 2D plane fundamentally changes this situation by resolving the kinetic energy spectrum into a ladder of discrete, yet highly degenerate, energy levels. In the absence of disorder these Landau levels (LL) have zero width, making electron-electron interactions impossible to ignore. At fields high enough that all electrons reside in the lowest ͑N 0͒ Landau level, interactions produce a plethora of incompressible quantized Hall phases at odd-denominator fractional fillings of the level, intriguing metallic phases at certain even-denominator fillings, and insulating states at very small fillings which presumably reflect pinned Wigner crystals  At lower magnetic fields, where the excited LLs are populated, a very different situation prevails. Recent experiments have shown that when three or more LLs are occupied (i.e., N $ 2) remarkable anisotropic collective states develop near half filling and curious isotropic insulating phases appear in the flanks of the LL  Between these two regimes lies the N 1 first excited LL. The discovery of an even-denominator fractional quantized Hall effect (FQHE) at Landau level filling fraction n 5͞2 gave the first indication that physics in the N 1 LL was different from that in the N 0 level  22 and a mobility of m ഠ 3.1 3 10 7 cm 2 ͞V s. The data in  The Hall resistance data at T ഠ 15 mK shown i

J P Eisenstein

K B Cooper

L N Pfeiffer

K W West

CiteSeerX

Insulating and fractional quantum hall states in the first excited landau level

The observation of new insulating phases of two-dimensional electrons in the first excited Landau level is reported. These states, which are manifested as reentrant integer quantized Hall effects, exist alongside well-developed even-denominator fractional quantized Hall states at ν = 7/2 and 5/2 and new odd-denominator states at ν = 3+1/5 and 3+4/5

Eisenstein, J. P.

Cooper, K. B.

Pfeiffer, L. N.

West, K. W.

Caltech Authors - Main

VOLUME 88, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 FEBRUARY 2002
076801Insulating and Fractional Quantum Hall States in the First Excited Landau Level
J. P. Eisenstein,1 K. B. Cooper,1 L. N. Pfeiffer,2 and K. W. West2
1California Institute of Technology, Pasadena, California 91125
2Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974
(Received 24 September 2001; published 30 January 2002)
The observation of new insulating phases of two-dimensional electrons in the first excited Landau
level is reported. These states, which are manifested as reentrant integer quantized Hall effects, exist
alongside well-developed even-denominator fractional quantized Hall states at n  72 and 52 and new
odd-denominator states at n  3 1 15 and 3 1
4
5 .
DOI: 10.1103/PhysRevLett.88.076801 PACS numbers: 73.43.Qt, 73.20.Qt, 73.63.HsAt zero magnetic field Coulomb interactions in two-
dimensional electron systems (2DES) are less impor-
tant than kinetic effects at all but the lowest densities. A
large magnetic field B applied perpendicular to the 2D
plane fundamentally changes this situation by resolving
the kinetic energy spectrum into a ladder of discrete, yet
highly degenerate, energy levels. In the absence of dis-
order these Landau levels (LL) have zero width, making
electron-electron interactions impossible to ignore. At
fields high enough that all electrons reside in the lowest
N  0 Landau level, interactions produce a plethora of
incompressible quantized Hall phases at odd-denominator
fractional fillings of the level, intriguing metallic phases at
certain even-denominator fillings, and insulating states at
very small fillings which presumably reflect pinned Wigner
crystals [1].
At lower magnetic fields, where the excited LLs are
populated, a very different situation prevails. Recent ex-
periments have shown that when three or more LLs are
occupied (i.e., N $ 2) remarkable anisotropic collective
states develop near half filling and curious isotropic insu-
lating phases appear in the flanks of the LL [2,3]. These
new states, which are quite distinct from the fractional
quantized Hall fluids found in the N  0 LL, are ob-
served only in very high quality samples and at very low
temperatures. Although their precise nature remains un-
clear, there is widespread belief that they are “stripe”
and “bubble” charge density wave (CDW) or liquid crys-
talline phases closely related to those originally predicted
by Hartree-Fock theory [4,5].
Between these two regimes lies the N  1 first excited
LL. The discovery of an even-denominator fractional
quantized Hall effect (FQHE) at Landau level filling frac-
tion n  52 gave the first indication that physics in the
N  1 LL was different from that in the N  0 level [6].
Since then only a small number of additional FQHE states
(at n  73, 83, and 72) have been observed. Most re-
cently it was reported that the 52 and 72 FQHE states are
replaced by strongly anisotropic phases when a substantial
magnetic field component Bjj parallel to the 2D plane
is added to the existing perpendicular field [7,8]. These
anisotropic phases appear to be similar to those found at-1 0031-90070288(7)076801(4)$20.00Bjj  0 in the N $ 2 LLs. Indeed, Rezayi and Haldane
have argued that the parallel field drives a transition from
a paired quantized Hall state to a stripe phase similar
to those found in the N $ 2 LLs [9]. Thus, the N  1
LL exhibits phenomena characteristic of both the N  0
FQHE and the N $ 2 CDW/liquid crystal regimes as well
as its own unique features. In this paper we report the
observation of several new phases in the N  1 LL. Most
importantly, eight isotropic insulating states exhibiting
integer Hall quantization have been found interposed
between various even- and odd-denominator fractional
quantized Hall states. These insulating states develop only
below T  50 mK.
Figure 1 shows magnetotransport data from a 2DES
sample consisting of a single 30 nm wide modulation-
doped GaAs quantum well. After brief low temperature il-
lumination with a red light emitting diode, the 2D electron
gas in this sample has a density of Ns  3.0 3 1011 cm22
and a mobility of m  3.1 3 107 cm2Vs. The data in
Fig. 1, which span the magnetic field range from B  3.2
to 4.0 T corresponding to Landau level filling factors be-
tween n  hNseB  4 and 3, reflect properties of the
2D system in the upper spin branch of the N  1 Landau
level. Both Hall Rxy and longitudinal Rxx data, taken
at our dilution refrigerator’s base temperature of about
T  15 mK and at T  50 mK, are shown.
The Hall resistance data at T  15 mK shown in Fig. 1a
contain several notable features. At the upper and lower
magnetic field extremes well-defined plateaus at Rxy 
h3e2 and h4e2, respectively, are evident. These plateaus
correspond to the conventional integer quantized Hall ef-
fects at Landau level filling factors n  3 and 4. Near
the center of the field range, at B  3.54 T, a flat Hall
plateau is apparent. The Hall resistance on this plateau
equals 2h7e2 to within 0.015%, thus verifying the ex-
istence of an even-denominator FQHE at n  72. Ac-
curately quantized Hall plateaus corresponding to FQHE
states at n  3 1 15 and 3 1
4
5 are also present. Interest-
ingly, the data offer only weak indications of FQHE states
at n  3 1 13 and 3 1
2
3 [10].
The most remarkable features in Fig. 1a are the four
isolated regions, denoted by arrows, in which the Hall© 2002 The American Physical Society 076801-1
VOLUME 88, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 FEBRUARY 2002800
400
0
R
xx
 
 
(O
hm
s)
4.03.83.63.43.2
Magnetic Field  (Tesla)
b)
0.35
0.30
0.25
R
xy
 
 
 
(h/
e2
)
h/4e2
h/3e2
7/2 3+1/5
3+4/5
a)
FIG. 1. (a) Hall resistance in the upper spin branch of the
N  1 Landau level. Solid curve, T  15 mK; dashed curve,
T  50 mK. Crosshairs indicate locations of FQHE states at
n  72, 3 1 15 , and 3 1
4
5 . Arrows denote locations of new
insulating states. (b) Longitudinal resistance data at the same
magnetic fields and temperatures.
resistance approaches either h3e2 or h4e2. The non-
monotonic variation of Rxy in these regions readily distin-
guishes them from the conventional n  3 and 4 integer
quantized Hall states. Of the four regions, the one
near B  3.78 T is the best developed; the data show
Rxy  h3e2 to within 0.02%. The regions near B  3.62
and 3.48 T, which straddle the n  72 FQHE state, are
clearly beginning to develop plateaus at Rxy  h3e2
and h4e2, respectively. The lowest field feature, near
B  3.35 T, is the weakest of the four, but it seems quite
likely that Rxy is approaching h4e2. These four features
are very temperature sensitive; the dashed trace in Fig. 1a
shows that they are completely absent at T  50 mK.
Figure 1b shows longitudinal resistance data from the
same field range as the Hall data in Fig. 1a. At B 
3.54 T a deep minimum in Rxx associated with the FQHE
state at n  72 is evident. The observed temperature de-
pendence of Rxx at this magnetic field is consistent with
thermal activation, Rxx ~ e2D2T ; we estimate D72 
70 mK. The Hall plateaus associated with the n  3 1 15
and 3 1 45 FQHE states are also accompanied by well-
defined minima in Rxx .076801-2The longitudinal resistance in the vicinity of each of the
four isolated regions where the Hall resistance approaches
either h3e2 or h4e2 exhibits somewhat intricate mag-
netic field and temperature dependences. These are illus-
trated in Fig. 2a where Rxx data taken at T  50, 40, 30,
and approximately 15 mK, are displayed. Concentrating
first on the narrow field range near B  3.78 T which is
expanded in Fig. 2b, the data show that a peak in Rxx devel-
ops as the temperature is reduced from T  50 to 40 mK.
By 30 mK this peak has grown substantially, shifted to
slightly lower magnetic field, and has been joined by a
weak relative minimum on its high field side. By approxi-
mately 15 mK the peak has grown and shifted slightly
more, but the minimum has evolved into a significant re-
gion of near zero longitudinal resistance. This zero in Rxx
coincides closely with the strong plateau in the Hall resis-
tance at Rxy  h3e2 noted previously.
The behavior of Rxx near the three other regions of reen-
trant Hall quantization is qualitatively similar, if less well
developed. For the two regions straddling the n  72
FQHE state, peaks in Rxx develop first upon cooling, but
these are eventually followed by the formation of adjacent
minima. The depth of these minima correlates well with
800
400
0
R
xx
 
 
(O
hm
s)
3.93.83.7
Magnetic Field  (Telsa)
b)
800
400
0
R
xx
 
 
(O
hm
s)
4.03.83.63.43.2
a)
7/2
FIG. 2. (a) Temperature dependence of the longitudinal re-
sistance in the upper spin branch of the N  1 Landau level
4 . n . 2. Light solid curve: 15 mK; dotted curve: 30 mK;
heavy solid curve: 40 mK; and dashed curve: 50 mK. (b) Ex-
panded view of Rxx associated with the reentrant integer Hall
quantized effect near B  3.78 T.076801-2
VOLUME 88, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 FEBRUARY 2002the strength of the developing plateaus in Rxy. For the
weakest feature, near B  3.35 T, a clear peak develops
and, at the lowest temperature, a shoulder on its low field
side suggests an incipient minimum.
The four regions of reentrant integer Hall quantization
are centered near Landau level filling factors n  3.28,
3.42, 3.56, and 3.70. The complex temperature depen-
dence of Rxx and integer quantization of Rxy around these
filling factors suggests that they are not quantum numbers
in the sense of the FQHE. Indeed, the minima in Rxx de-
scribed above are associated with the onset of insulating
behavior, not the formation of new FQHE states.
Figure 3 demonstrates that similar reentrant integer
quantized Hall effect (RIQHE) states also appear in the
lower spin branch of the N  1 LL, between filling
factors n  2 and 3. In these data both of the reentrant
Hall features above half filling exhibit strong plateaus
accurately quantized to h2e2. Anomalous minima and
maxima in Rxy in this filling factor range have been
reported before [11], but the present data are, to our
knowledge, the first to demonstrate reentrant integer Hall
quantization. Pan et al. [11] also observed large changes
in the anomalous Rxy features when the direction of the
perpendicular magnetic field was reversed. This was at-
tributed to mixing of the longitudinal and Hall resistances
in the sample. We emphasize that in our sample field
reversal has a negligible effect. Finally, a strong n  52
fractional quantized Hall effect is also apparent in Fig. 3.
The temperature dependence of Rxx at n  52 suggests
an energy gap D52  310 mK.
The transport data described above were obtained with
the magnetic field perpendicular to the 2D plane. Exami-
nation of the longitudinal and Hall resistances for sensi-
tivity to the direction of current flow within the 2D plane
shows no evidence, in the N  1 LL, of the large sys-
tematic anisotropy which is readily apparent in the N $ 2
levels. On the other hand, it is known that the imposi-
0.50
0.45
0.40
0.35
R
xy
 
 
 
 
(h/
e2
)
5.55.04.5
Magnetic Field  (Tesla)
h/2e2
h/3e2
5/2
FIG. 3. Hall resistance data in the lower spin branch of the
N  1 Landau level, 3 . n . 2, at T  15 mK. New insulat-
ing states are indicated by arrows, crosshairs locate the n  52
FQHE state.076801-3tion of a magnetic field component Bjj in the 2D plane
destabilizes the n  52 and 72 FQHE states [12] and
renders the longitudinal resistance of the sample strongly
anisotropic [7,8,13]. By tilting the present sample relative
to the total applied magnetic field we have corroborated
these earlier findings and have found that the new reentrant
integer quantized Hall features reported here are gradually
suppressed as the in-plane field is increased.
The discovery of reentrant integer quantized Hall states
in the N  1 Landau level suggests the existence of
collective insulating states. If the electrons in the partially
filled uppermost LL are insulating, the filled LLs beneath
the Fermi level ensure integer quantization of the Hall
resistance and zero longitudinal resistance. The fact that
these new insulators are reentrant, i.e., are separated from
one another and the conventional integer QHE states by
magnetic field ranges exhibiting fractional quantized Hall
states or simply finite Rxx and nonquantized Rxy , is strong
evidence that the origin of the insulating behavior is collec-
tive. Even if the conventional integer QHE states at n  3
and n  4 are dominated by disorder-induced single-
particle localization, the presence of conducting regions
between them and the reentrant insulating states neces-
sarily implies that the quasiparticles in the system first
delocalize upon moving away from integer filling. It seems
highly unlikely that the same single-particle mechanism
could then become operative again at the much higher
quasiparticle densities present in the reentrant insulating
states. Furthermore, the existence of the FQHE states at
n  3 1 15 and n  3 1
4
5 offers compelling evidence
that electron-electron interactions are already dominant
even before the first reentrant insulator has formed.
Of the four reentrant IQHE states observed in the up-
per spin branch of the N  1 LL (i.e., 4 . n . 3), only
the one near B  3.78 T actually exhibits vanishing Rxx
at T . 15 mK. As suggested above, we believe that the
nonzero Rxx of the other three states merely reflects their
weaker development. In each of the four cases, a peak in
Rxx develops first upon cooling and is followed by the for-
mation of a minimum at lower temperatures. We speculate
that this behavior reflects the shifting balance of current be-
tween the edge channels of the filled lower LL levels and
the “valence” electrons in the uppermost LL. If the valence
electrons are highly conducting, say, at high temperature,
then Rxx will obviously be small. On the other hand, if the
valence electrons are insulating, the resistance Rxx will be
essentially zero owing to the edge channels of the LLs be-
neath the Fermi level. A peak in Rxx is therefore expected
at some intermediate value of the longitudinal conductivity
of the valence electrons.
The reentrant insulating states reported here are remi-
niscent of the reentrant integer QHE states recently found
in the N $ 2 LLs [2,3]. Two such insulating states, one
near 14 filling and the other near 34 filling, are observed
within each spin sublevel. There is evidence, both experi-
mental [14] and theoretical [15,16], that these states are076801-3
VOLUME 88, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 FEBRUARY 2002related to the bubble charge density waves predicted by
Koulakov, Fogler, and Shklovskii [4] and by Moessner and
Chalker [5] on the basis of Hartree-Fock theory. The bub-
ble phases, which consist of triangular arrays of multielec-
tron clusters, are interposed between unidirectional stripe
phases around half filling and Wigner crystals deep in the
flanks of the level. Pinning of the bubble arrays presum-
ably produces the observed insulating behavior. We note,
however, that the theoretical work on bubble phases has so
far been aimed at the N $ 2 LLs, not the N  1 level.
Reentrant insulating states have also been observed in
the N  0 lowest LL [17]. For example, the Laughlin
liquid FQHE state at n  15 is flanked by insulating be-
havior above and in a narrow window below it in magnetic
field [18]. A plausible interpretation [19] of this is that
the insulating behavior results from the pinning of mag-
netic field-induced Wigner crystals, but reentrance occurs
because the downward cusp in the energy of the Laugh-
lin liquid makes it the ground state in a narrow range of
filling factor around n  15. For two-dimensional hole
systems, a similar reentrance is observed [20] around the
n  13 FQHE state. The shift of the reentrance to higher
filling factors in 2D hole systems has been attributed to the
increased importance of Landau level mixing.
At present it is not possible to determine whether the
new insulating states in the N  1 LL reported here bear
close resemblance to either the anticipated bubble phases in
the N $ 2 LLs or the conventional Wigner crystals in the
N  0 level. A distinguishing feature of the present case
is that there are four reentrant integer quantized Hall states
in each spin branch of the N  1 level, two above and two
below half filling. In each spin sublevel the innermost pair
of these states occur quite close to half filling. If these are,
in fact, Wigner crystals, they are occurring at unusually
high densities of the carriers in the valence Landau level.
In conclusion, a number of new insulating electronic
phases have been observed in the first excited Landau level
of ultraclean 2D electron systems. The origin of the insu-
lating behavior is quite likely a collective effect, but in the
absence of theoretical predictions a detailed understanding
is not yet possible. We mention two intriguing preliminary
findings which may assist in developing this understand-
ing. First, the dc current-voltage characteristics of the in-
sulating state near B  3.78 T exhibit sharp and hysteretic
thresholds to conduction that may reflect depinning [14].
Second, the insulating state near B  5.1 T in the lower076801-4spin branch of the N  1 LL exhibits small amounts of
hysteresis with respect to the direction of magnetic field
sweep, possibly suggesting a first order phase transition.
This work was supported by the DOE under Grant
No. DE-FG03-99ER45766 and the NSF under Grant
No. DMR0070890.
[1] For a review, see Perspectives in Quantum Hall Effects,
edited by S. Das Sarma and A. Pinczuk (John Wiley, New
York, 1997).
[2] M. P. Lilly et al., Phys. Rev. Lett. 82, 394 (1999).
[3] R. Du et al., Solid State Commun. 109, 389 (1999).
[4] A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys.
Rev. Lett. 76, 499 (1996); M. M. Fogler, A. A. Koulakov,
and B. I. Shklovskii, Phys. Rev. B 54, 1853 (1996); M. M.
Fogler and A. A. Koulakov, Phys. Rev. B 55, 9326 (1997).
[5] R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006
(1996).
[6] R. L. Willett et al., Phys. Rev. Lett. 59, 1776 (1987).
[7] W. Pan et al., Phys. Rev. Lett. 83, 820 (1999).
[8] M. P. Lilly et al., Phys. Rev. Lett. 83, 824 (1999).
[9] E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 84,
4685 (2000).
[10] There are theoretical suggestions that the “fifths” will be
stronger than the “thirds” in the N  1 Landau level. See
A. H. MacDonald and S. M. Girvin, Phys. Rev. B 33, 4009
(1986); N. d’Ambrumenil and A. M. Reynolds, J. Phys. C
21, 119 (1988).
[11] W. Pan et al., Phys. Rev. Lett. 83, 3530 (1999).
[12] J. P. Eisenstein et al., Phys. Rev. Lett. 61, 997 (1988); Surf.
Sci. 229, 31 (1990).
[13] The “hard” transport direction is parallel to the in-plane
magnetic field.
[14] K. B. Cooper et al., Phys. Rev. B 60, R11 285 (1999).
[15] F. D. M. Haldane, E. H. Rezayi, and Kun Yang, Phys. Rev.
Lett. 85, 5396 (2000).
[16] N. Shibata and D. Yoshioka, Phys. Rev. Lett. 86, 5755
(2001); D. Yoshioka and N. Shibata, cond-mat/0106099.
[17] See the chapter by M. Shayegan in Ref. [1] for review of
insulating states in the N  0 LL.
[18] When n , 12 there are no Landau levels beneath the
Fermi level, and thus insulating electronic configurations
display divergent, as opposed to vanishing, longitudinal
resistance.
[19] H.-W. Jiang et al., Phys. Rev. Lett. 65, 633 (1990).
[20] H. C. Manoharan and M. Shayegan, Phys. Rev. B 50,
17 662 (1994).076801-4


Insulating and Fractional Quantum Hall States in the First Excited Landau Level

Crossref