VOLUME
82, NUMBER 2 PHYSICAL REVIEW LETTERS 11J
ANUARY
1999
Evidence for an Anisotropic State of Two-Dimensional Electrons in High Landau Levels
M. P. Lilly,
1
K.B. Cooper,
1
J.P. Eisenstein,
1
L. N. Pfeiffer,
2
and K.W. West
2
1
California Institute of Technology, Pasadena, California 91125
2
Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974
(Received 19 August 1998)
Magnetotransport experiments on high mobility two-dimensional electron gases in GaAsyAlGaAs
heterostructures have revealed striking anomalies near half filling of several spin-resolved, yet
highly excited, Landau levels. These anomalies include strong anisotropies and nonlinearities of the
longitudinal resistivity r
xx
which commence only below about 150 mK. These phenomena are not
seen in the ground state or first excited Landau level but begin abruptly in the third level. Although
their origin remains unclear, we speculate that they reflect the spontaneous development of a generic
anisotropic many-electron state. [S0031-9007(98)08178-2]
PACS numbers: 73.20.Dx, 73.40.Kp, 73.50.Jt
A magnetic field applied perpendicular to the plane of a
two-dimensional electron gas (2DEG) resolves the energy
spectrum into discrete Landau levels (LLs). As the field
increases, the Fermi level drops down through the Landau
ladder in a series of steps until, at high field, it resides in the
lowest (N 0) level. In this situation the kinetic energy
of the electrons is quenched and electron-electron interac-
tions dominate the physics with the fractional quantized
Hall effect (FQHE) as the most spectacular consequence
[1]. After more than 15 years of study, much is known
about electron correlations in this lowest LL case. The
same cannot be said when the Fermi level is in a higher
Landau level. In the second LL (N 1), the FQHE is vir-
tually absent; only fragile and poorly understood states at
Landau filling fractions n 7y3,5y2, and 8y3 are seen in
the best samples. In the third and higher LLs (N $ 2) still
less is known, although there have been interesting sug-
gestions of charge density waves in the clean limit [2,3].
At very high N, and therefore very low magnetic field, the
Landau level splitting becomes insignificant and the 2DEG
assumes the character of a weakly disordered Fermi liquid.
In this paper we report the observation of several
dramatic anomalies in the low temperature magnetotrans-
port of clean 2DEGs when the Fermi level lies near the
middle of a spin-resolved highly excited Landau level.
These effects, which commence only below about
150 mK, abruptly begin and are strongest in the third
(N 2) LL, but persist up to about N 6. Including
strong anisotropies and intriguing nonlinearities of the
resistivity, these effects suggest a considerably more
interesting tableau at high N than independent electrons
moving in a disordered Landau band.
The samples used in this study are GaAsyAlGaAs
heterojunctions grown by molecular beam epitaxy (MBE).
Data from six samples (A through F) will be discussed.
Samples A, B, and C were taken from one MBE wafer, D
and E from a second, and F from a third. Each wafer
was rotated during growth to ensure high homogeneity
of the electron density n
s
. These densities (in units of
10
11
cm
22
) are close to n
s
2.67 for samples A, B, and
C; n
s
2.27 for samples D and E; and n
s
1.52 for
sample F. The low temperature mobility of each is m$
9310
6
cm
2
yV s. Each sample was cleaved (along k110l
directions) into a 5 3 5 mm square from its parent k001l
wafer. For samples A, D, E, and F, indium contacts were
placed at the corners and the midpoints of the sides of
the chip. Hall bar patterns were lithographically etched
onto samples B and C before the contacts were made.
The samples were briefly illuminated at low temperature
with a red LED. Electrical transport measurements were
performed using 2–20 nA, 13 Hz excitation, although
for the nonlinearity studies an additional dc current was
imposed.
Figure 1 shows the resistivity [4] r
xx
at T 150 mk
of sample A. Shubnikov-deHaas (SdH) oscillations com-
mence at around B ø 60 mT and the spin splitting of the
Landau levels is evident by 130 mT. The smallness of
these fields attests to the high quality of the 2DEG in
this sample. Between about B 5.5 and 11 T the Fermi
level is in the upper spin branch of the N 0 lowest
LL. This corresponds to Landau level filling fractions
n hn
s
yeB between n 2 and n 1. Clear signatures
0 2 4 6 8 10
0
100
200
300
B (Tesla)
ρ
xx
(Ω)
N=0N=123
...
5/2
ν=2
5/3 4/3
T=150mK
N=2 Landau level
11/2
9/2
FIG. 1. Overview of diagonal resistivity in sample A at T
150 mk. Structure in the N 2 Landau level is expanded in
the inset.
394 0031-9007y99y82(2)y394(4)$15.00 © 1999 The American Physical Society
VOLUME
82, NUMBER 2 PHYSICAL REVIEW LETTERS 11J
ANUARY
1999
1.5 2 2.5 3 3.5
0
250
500
750
1000
B (Tesla)
ρ
xx
(Ω)
N=1N=2N=3
...
15/2
13/2
11/2
9/2
7/2
25 100
0
1000
T (mK)
ρ
xx
FIG. 2. Peaks in r
xx
in sample A developing at low tem-
peratures in high LLs (dotted line: T 100 mK; thick line:
65 mK; thin line: 25 mK). Inset: temperature dependence of
peak height at n 9y2 (closed circles), 11y2 (open circles),
13y2 (closed triangles), and 15y2 (open triangles).
of several fractional quantized Hall states are evident. Be-
tween B 2.8 and 5.5 T the Fermi level is in the N 1
second LL and the developing n 5y2 FQHE is indicated
in the figure. At lower temperatures this state, and the
nearby n 7y3 state, strengthen. In the upper spin branch
of this same LL a minimum in r
xx
is seen at n 7y2, but
it does not deepen as the temperature is reduced.
Below B ø 2.8 T the Fermi level is in the N 2 and
higher LLs. The inset to Fig. 1 reveals that in both spin
branches of the N 2 LL there are a number of maxima
and minima in r
xx
. Although this complex structure does
not appear to be associated with a FQHE (no quantized
plateaus in the Hall resistivity r
xy
have been found),
its existence strongly indicates that electron correlations
are important. In the standard model [5], if disorder
overwhelms interactions, a simple peak in the resistivity
separates the broad zeros of adjacent integer quantum Hall
states.
Figure 2 displays the temperature dependence of the
r
xx
features in the N 1, N 2, and N 3 LLs in
sample A. Below T 150 mk peaks develop at n
9y2,11y2, 13y2, and 15y2 which grow rapidly below
100 mK; by 25 mK the peak at n 9y2 has exceeded
1000 V. Surprisingly, the peaks do not narrow as T is
reduced. The subsidiary structures flanking the peaks do
fall with temperature but are not simply “consumed” by
the widening of the nearby integer QHE states; even at
25 mK they are still evident [6]. This behavior does not
fit the standard model of disorder-driven integer QHE
transitions, but suggests instead that interactions remain
important down to very low T . The observed behavior
of r
xx
in the N 1 LL (only the upper spin branch, 4 .
n.3, is shown in the figure) is qualitatively different.
Instead of peaks, there are minima at n 7y2 and at the
fragile fractional QHE state at n 5y2. The closeness
in magnetic field of the n 9y2 and 7y2 filling factors
makes this difference particularly striking. This basic
0 1 2 3 4 5
0
250
500
750
1000
B (Tesla)
ρ
xx
(Ω)
N=123
...
IV
I
V
5/27/2
9/2
11/2
13/2
T=25mK
FIG. 3. Anisotropy of r
xx
in sample A at T 25 mK. The
two traces result from simply changing the direction of current
through the sample; the sample itself is not rotated.
result is just as clearly evident in samples B through F.
In the inset to Fig. 2, r
xx
at n 9y2,11y2, 13y2, and
15y2 is plotted versus temperature.
Figure 3 displays our most remarkable finding. The
two traces are resistances measured in sample A at T
25 mk for two perpendicular directions of the current
flow through the sample. The diagrams in the figure
depict the difference between the two configurations. This
seemingly innocuous change vastly alters the resistivity
in the N 2 and several higher LLs. Note that the
solid curve has been multiplied by a factor of 0.62 in
order to match the data sets in the very low field regime.
A variation of this size in the resistance of a quantum
Hall sample is quite common and may simply reflect
irregularities in the positions of the contacts. In any case,
this factor in no way obscures the dramatic anisotropy
of the resistivity near the centers of both spin branches
of the N 2 through N 5 LLs. While peaks in r
xx
are evident in one configuration, relatively deep minima
are seen in the other [7]. At n 9y2 the ratio of the
resistances is close to 100. Equally striking is the fact
that no comparable anisotropy is apparent in the N
1 LL (nor the N 0 level which is not shown in the
figure). The anisotropy we are reporting is apparently
confined to the centers of the LLs; well away from half
filling (of each spin branch) the two resistances again
roughly match (after the scaling factor of 0.62 is applied).
As the temperature is increased, the anisotropy in r
xx
subsides until by T 150 mK it is no longer significant.
We emphasize that for no current configuration are any
plateaus seen in the Hall resistance r
xy
at half filling of
the N $ 2 Landau levels.
By measuring r
xx
with various contact configurations
we have determined that the orientation of the anisotropy
is fixed within the sample and is insensitive to reversal
of the magnetic field and thermal cycling to room tem-
perature. Although the geometry of sample A is quite
open and the precise current distribution is unknown, our
data suggest that the “principal axes” of the anisotropy are
395
VOLUME
82, NUMBER 2 PHYSICAL REVIEW LETTERS 11J
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1999
0 1 2 3
0
500
1000
B (Tesla)
ρ
xx
(Ω)
a)
9/2
7/2
HALL BARS
V
V
1.3 1.4 1.5
b)
9/2
SAMPLE F
FIG. 4. (a) Anisotropy of r
xx
at T 25 mK measured
using Hall bar samples B (dashed line) and C (solid line).
(b) Anisotropy of r
xx
at T 15 mK measured in the low
density sample F. Current flow configurations as in Fig. 3.
roughly parallel to the sides of the chip. In order to better
define the current path, Hall bar samples B and C were
examined. In sample B the bar axis was oriented paral-
lel to the k1
10l crystal axis, while in sample C the bar
was aligned along k110l. Figure 4a compares r
xx
data
from these two samples. At n 9y2 the two bars ex-
hibit a large (roughly sixfold) anisotropy in r
xx
while at
n 7y2 in the N 1 LL the observed r
xx
is isotropic.
(The data from Hall bar C were multiplied by a factor of
0.75 to match the amplitude of its SdH oscillations at very
low field with those of Hall bar B.) Though not shown
in the figure, isotropy is also observed at n 5y2 and
3y2. As in sample A, substantial anisotropy exists near
n 11y2,13y2, 15y2, and 17y2 before isotropy returns
at low magnetic field. The orientation of the anisotropy in
sample A agrees with that seen in the Hall bars. While the
magnitude of the n 9y2 anisotropy in the Hall bars is
weaker than that seen in the sample A, it seems clear that
the same basic effect is at work. Finally, the anisotropy
effect is also seen in the lower density samples D and
F, which come from different MBE wafers. Figure 4b
shows data from sample F in which the 9y2 anisotropy
is nearly 100-fold even after magnifying the data from the
r
xx
-minimum configuration by a factor of 1.6 to match
the low field SdH oscillations.
The sudden development at very low temperatures of
strong temperature dependences and large anisotropies of
the resistivity of clean 2DEGs in the third and several
higher Landau levels suggests that some previously unap-
preciated physics is at work. The anisotropy is intriguing
as it is not at all clear what breaks the in-plane symme-
try of the system. There are various extrinsic effects re-
lated to the MBE growth that might pick out a direction
in the 2D plane. A wafer-scale gradient in the electron
density due to the off-normal positions of the various ele-
mental sources in the MBE chamber is one possibility.
However, our samples were rotated during their growth
specifically to minimize such gradients. Indeed, measure-
ments of the SdH periodicity using many different voltage
and current configurations (including those which produce
the large anisotropy displayed in Fig. 3) show no more than
a 0.3% variation. This result strongly suggests that there
is at most a very small density gradient. At the same time,
the low onset field of the SdH oscillations (B ø 60 mT,
corresponding to Landau filling fraction n ø 180) seen
in all contact configurations suggests that density fluctua-
tions on short distance scales are also small. Among other
possible extrinsic symmetry-breaking effects, we mention
the often observed “slip lines” which are believed to be
steps on the surface of MBE wafers, and the possibility
that the k001l GaAs substrate was slightly miscut from its
parent boule.
It is possible that the observed anisotropy in r
xx
is due to some unforeseen order in the static disorder
potential. This is unlikely since the anisotropy is seen
only in the N 2 and a handful of higher LLs. It is not
present in the lowest or first excited LL, nor is it seen
in the semiclassical regime well below B 1 T. The
effects reported here are, however, reminiscent of recently
observed transport anisotropies near n 1y2 in the
lowest LL in samples where a periodic density modulation
has been externally imposed [8,9]. A theoretical basis
for understanding this effect has been proposed [10],
and it may turn out to be useful in the present context
[11], although it would not identify the source of the
inhomogeneity.
Recently, Koulakov, Fogler, and Shklovskii [2] and
subsequently Moessner and Chalker [3] have proposed
that in a clean 2DEG in the N 2 and higher LLs
the uniform electron liquid may be unstable against the
formation of charge density waves (CDW). They further
suggest that near half filling of the LL the CDW is a
unidirectional “stripe phase” having a wavelength of order
the cyclotron radius. In this stripe phase the electron
density in the uppermost LL alternates between zero and
full filling. At n 9y2 this implies there are stripes of
the incompressible QHE states n 4 and n 5. While
it is surely plausible that electrical transport in such a
unidirectional phase would be anisotropic, it is not clear
what would pin the stripes or why they are apparently
coherent over the macroscopic size of our samples.
To further investigate magnetotransport in highly ex-
cited LLs, we have examined the linearity of r
xx
. Fig-
ure 5 summarizes our T 25 mK results from sample E.
The figure displays the differential resistivity dV
xx
ydI at
n 9y2,11y2, etc., measured using 5 nA, 13 Hz excita-
tion, in the presence of an added dc current I
dc
. The data
were taken using a contact configuration for which the re-
sistivity (i.e., dV
xx
ydI at I
dc
0) exhibits strong peaks
at these filling factors. In each panel I
dc
runs from 2200
to 1200 nA, and the plotted dV
xx
ydI data are normal-
ized by their value at I
dc
0. Several of the data sets
show marked nonlinearity. At n 9y2, where the effect
is strongest, dV
xx
ydI at first increases substantially as I
dc
is applied. At about 6100 nA it reaches a maximum and
then falls off at higher current. Since r
xx
at n 9y2
falls with rising temperature, it is clear that the initial rise
in dV
xx
ydI at small I
dc
is not consistent with electron
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VOLUME
82, NUMBER 2 PHYSICAL REVIEW LETTERS 11J
ANUARY
1999
9/2 11/2
13/2
normalized dV /dI
xx
15/2
17/2 19/2
-200 0 200
0.5
1
1.5
21/2
I
dc
(nA)
23/2
FIG. 5. Nonlinearity of differential resistivity dV
xx
ydI in
sample C at half filling of several high LLs at T 25 mK.
The resistivity is normalized by its value at I
dc
0. In each
panel I
dc
runs from 2200 to 1200 nA and the normalized
dV
xx
ydI from 0.5 to 1.5.
heating. At large I
dc
the falling of dV
xx
ydI may indeed
be due to heating. Qualitatively similar nonlinearities are
seen at half filling of the higher LLs, but the effect eventu-
ally dies away at low magnetic field. Note the oscillatory
strength of the nonlinearity: strong at 9y2, weaker at 11y2,
relatively stronger at 13y2, weak again at 15y2, and so on.
As with the anisotropy, the nonlinearity is a low tempera-
ture phenomenon; by 150 mK it is essentially gone. It
is more fragile than the anisotropy; while clearly evident
in some samples it is weaker in others. We emphasize
that in each LL the nonlinearity is seen over a range of n
around half filling. Finally, although weak nonlinearities
have sometimes been seen in the N 1 and N 0 LL,
they appear qualitatively different from that seen in the
N 2 and higher LLs.
The increasing differential resistivity for small dc
currents also implies an increasing conductivity. This
follows from inverting the resistivity tensor and noting
that the Hall resistivity r
xy
is several times larger than r
xx
over the field range of interest. Although an increasing
conductivity is suggestive of a depinning mechanism,
we have seen no evidence of a sharp threshold; the
nonlinearity appears to turn on continuously.
In conclusion, we have reported dramatic magneto-
transport anomalies that are specific to the third and higher
Landau levels in clean 2DEG’s in GaAsyAlGaAs het-
erostructures. These effects, which appear only at low
temperature, include huge anisotropies in the resistivity
near half filling of the spin-resolved Landau levels as well
as intriguing nonlinearities. The origin of these effects is
unclear, but the data are consistent with the spontaneous
development of an anisotropic electronic configuration.
The occurrence of the same basic transport signatures in
several adjacent highly excited Landau levels points to a
generic mechanism.
We thank J. Chalker, S. M. Girvin, B. Halperin, A. H.
MacDonald, and K. Yang for helpful discussions and Ian
Spielman for help in determining the crystallographic axes
in our samples. This work was supported by the National
Science Foundation.
[1] For recent reviews, see Perspectives in Quantum Hall
Effects, edited by S. Das Sarma and A. Pinczuk (John
Wiley, New York, 1997).
[2] A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys.
Rev. Lett. 76, 499 (1996); Phys. Rev. B 54, 1853 (1996);
M.M. Fogler and A. A. Koulakov, Phys. Rev. B 55, 9326
(1997).
[3] R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006
(1996).
[4] As is commonplace, we employ the term resistivity even
though the actual measured quantity is resistance; in
square samples the factor relating the two is of order unity.
[5] See, for example, the chapter by S. Das Sarma in Ref. [1].
[6] At very low T we find tiny peaks in r
xx
separated from
the main peak at n 9y2 and n 11y2 by narrow
regions in which r
xx
ø 0. We find that in these regions
the Hall resistance is quantized, but at the value of the
nearby integer QHE.
[7] In the N 2 LL, this observation was, to our knowledge,
first made by R. L. Willett and J. P. Eisenstein (unpub-
lished) and has also been reported by H. L. Stormer et al.,
Bull. Am. Phys. Soc. 38, 235 (1993).
[8] R. L. Willett, K. W. West, and L. N. Pfeiffer, Phys. Rev.
Lett. 78, 4478 (1997).
[9] J. Smet, K. von Klitzing, D. Weiss, and W. Wegscheider,
Phys. Rev. Lett. 80, 4538 (1998).
[10] Felix von Oppen, Ady Stern, and Bertrand I. Halperin,
Phys. Rev. Lett. 80, 4494 (1998).
[11] B. Halperin (private communication).
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