Frictional drag measurements revealing anomalously large dissipation at the
transition between the weakly- and strongly-coupled regimes of a bilayer
two-dimensional electron system at total Landau level filling factor $\nu_T =1$
are reported. This result suggests the existence of fluctuations, either static
or dynamic, near the phase boundary separating the quantized Hall state at
small layer separations from the compressible state at larger separations.
Interestingly, the anomalies in drag seem to persist to larger layer
separations than does interlayer phase coherence as detected in tunneling.Comment: 4 pages, 4 figure

Eisenstein, J. P.

Kellogg, M.

Pfeiffer, L. N.

West, K. W.

English

arXiv

Frictional drag measurements revealing anomalously large dissipation at the transition between the weakly- and strongly-coupled regimes of a bilayer two-dimensional electron system at total Landau level filling factor ν_T = 1 are reported. This result suggests the existence of fluctuations, either static or dynamic, near the phase boundary separating the quantized Hall state at small layer separations from the compressible state at larger separations. Interestingly, the anomalies in drag seem to persist to larger layer separations than does interlayer phase coherence as detected in tunneling

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Double layer two-dimensional electron systems: Probing the transition from weak to
strong coupling with Coulomb drag
M. Kellogg1, J. P. Eisenstein1, L. N. Pfeiffer2, and K. W. West2
1California Institute of Technology, Pasadena CA 91125
2Bell Laboratories, Lucent Technologies,
Murray Hill, NJ 07974
Frictional drag measurements revealing anomalously large dissipation at the transition between
the weakly- and strongly-coupled regimes of a bilayer two-dimensional electron system at total
Landau level filling factor νT = 1 are reported. This result suggests the existence of fluctuations,
either static or dynamic, near the phase boundary separating the quantized Hall state at small
layer separations from the compressible state at larger separations. Interestingly, the anomalies in
drag seem to persist to larger layer separations than does interlayer phase coherence as detected in
tunneling.
At high magnetic fields two-dimensional electron sys-
tems (2DES) exhibit a variety of collective states. For
example, if the perpendicular magnetic field is adjusted
so that the density of electrons Ns equals one-half the de-
generacy eB/h of the lowest spin resolved Landau level
(i.e. at filling factor ν = Nsh/eB = 1/2), the resulting
strongly correlated electron system can be successfully
modeled as a metallic liquid of composite fermions[1].
The system crudely resembles a conventional 2DES in
zero magnetic field. No quantized Hall effect is seen.
Remarkably, the system possesses a well-defined Fermi
surface and quasiparticles which move in semiclassical
cyclotron orbits whose diameters diverge as ν → 1/2.
Now consider a system consisting of two parallel
2DESs, each at ν = 1/2, separated by a barrier layer.
Obviously, if the separation d between the two 2DESs
is large, they are uncoupled and the net system behaves
much the same as a single layer. In contrast, if d is very
small (less than the average separation between electrons
in each layer) the ground state of the bilayer system is
qualitatively different. In this strongly-coupled limit in-
terlayer Coulomb interactions engender a novel broken
symmetry, spontaneous interlayer phase coherence, in
which all electrons are coherently spread between both
layers, even in the hypothetical limit of zero interlayer
tunneling[2]. This bizarre state, which is best character-
ized by the total filling factor νT = 1, may be viewed
in a number of equivalent ways, including as an itiner-
ant pseudospin ferromagnet or a Bose condensate of in-
terlayer excitons. Experimentally, the system has been
found to display numerous striking properties, including
the quantized Hall effect (at Rxy = h/e
2)[3], a pseu-
dospin textural phase transition[4, 5], Josephson-like in-
terlayer tunneling[6], and, most relevant here, quantized
Hall drag[7]. Additional properties, including counter-
flow superfluidity, are anticipated[8].
The nature of the transition between the strongly-
coupled ferromagnetic or excitonic phase at small d and
the weakly-coupled composite fermion liquids at large
d is very poorly understood. As d increases from zero
(at zero temperature) the ferromagnetic phase suffers in-
creasingly severe quantum fluctuations which eventually
destroy the (algebraic) pseudomagnetic order at a crit-
ical layer separation. Although there exists numerical
evidence suggesting that this quantum phase transition
may be weakly first order[9], current experiments suggest
a continuous transition.
Beyond the nature of the demise of the ferromagnetic
phase as d increases, remains the question of the ground
state of the system above the critical separation. Be-
fore reverting to independent composite fermion liquids
at very large d, there is the possibility of additional inter-
layer correlated phases at intermediate separations. Can-
didate states include bilayer Wigner crystals[10], paired
composite fermion liquids[11], and various other exotic
phases which may share some but not all of the prop-
erties of the ferromagnetic phase (e.g. interlayer phase
coherence but no quantized Hall effect, etc.)[12, 13]. In
this paper we report interlayer friction measurements
(“Coulomb drag”) which shed light on the transition
between the strongly- and weakly-coupled regimes at
νT = 1. In particular, we observe a large enhancement
of the longitudinal component of the drag near the tran-
sition. Although rapidly attenuated, this enhancement
persists to surprisingly large layer separations.
Drag measurements are performed by driving a current
I, typically 1nA at 5Hz, through one of the two layers
of a double layer 2DES while monitoring the voltage VD
which appears in the other, electrically isolated, layer.
At zero magnetic field the drag resistance RD = VD/I
provides a unique measure of the interlayer momentum
relaxation rate. For closely-spaced layers and low tem-
peratures this rate is dominated by simple Coulomb scat-
tering and hence the moniker Coulomb drag.
The samples used in the present experiments consist of
two 18nm GaAs quantum wells separated by a 10nm bar-
rier layer of Al0.9Ga0.1As. Each quantum well contains a
2DES which, in the sample’s as-grown state, has a den-
sity of aboutNs ≈ 5.3×10
10cm−2 and a low-temperature
mobility of µ ≈ 106cm2/Vs. Data from three samples, A,
2FIG. 1: Coulomb drag resistances in sample A at T = 30mK,
in the vicinity of νT = 1. Left column: Hall drag Rxy,D; right
column: longitudinal drag Rxx,D. Each row corresponds to a
different 2D density in the bilayer sample, and thus a different
effective layer separation d/ℓ at νT = 1.
B, and C, are reported here. Samples A and B consist
of square mesas, 250µm on a side, with four arms ex-
tending outward to remote ohmic contacts. Sample C is
bar-shaped, 40 × 400µm, and has five arms and ohmic
contacts. These contacts are connected to one or the
other 2DES using a selective depletion scheme[14]. The
densities of the individual 2DESs in the central mesa re-
gion are controlled with electrostatic gates deposited on
the top and backside of the samples. A detailed study
of the Coulomb drag in these samples at zero magnetic
field has been reported elsewhere[15].
Figure 1 shows representative Coulomb drag data from
sample A in the vicinity of νT = 1 at T = 30mK. Each
row corresponds to a different effective layer separation
d/ℓ, with d = 28nm the center-to-center quantum well
separation and ℓ = (h¯/eB)1/2 the magnetic length at
νT = 1. This key ratio governs the relative importance
of inter- and intra-layer Coulomb interactions in the bi-
layer system, and can be varied in situ by symmetrically
changing the electron densities in the individual quan-
tum wells. The data in Fig. 1 is therefore plotted versus
the inverse total filling factor νT
−1, instead of magnetic
field, to aid in comparing the different densities. The
right-hand panels display the longitudinal drag resistance
Rxx,D (i.e. drag voltage parallel to the current flow in
the drive layer) while the left-hand panels present the
transverse, or Hall, drag resistance Rxy,D.
The top two panels (a and e) of Fig. 1 show Hall and
longitudinal drag data at d/ℓ = 1.60. At this effective
layer separation the system is well within the strongly-
coupled bilayer quantum Hall effect phase and, as re-
ported previously[7], at νT = 1 the Hall drag exhibits a
plateau accurately quantized at Rxy,D = h/e
2 while the
longitudinal drag Rxx,D is essentially zero. While it is
not surprising that Rxx,D is zero in the quantized Hall
state (the gap to charged excitations suppressing dissipa-
tion at low temperatures), the quantization of Hall drag
is a dramatic consequence of the physics of the strong
coupling regime and supports, albeit indirectly, the exis-
tence of counterflow superfluidity[8].
On moving away from νT = 1, the Hall drag rapidly di-
minishes while the longitudinal drag displays two strong
maxima. The sign of the Hall drag voltage is the same
as that of the conventional Hall voltage in the current-
carrying layer. In contrast, the sign of the longitudinal
drag voltage is the opposite of the conventional longitudi-
nal voltage drop in the current-carrying layer. Indeed, for
all the data reported here the sign of Rxx,D is the same
as that encountered at zero magnetic field where the drag
voltage reflects the electric field needed to counteract the
frictional force due to the current flow in the drive layer.
The remaining panels of Fig. 1 show how Coulomb
drag changes as d/ℓ increases and the strongly-coupled
bilayer quantized Hall phase collapses. Panels b, c, and d
show that the Hall drag plateau becomes a rapidly sub-
siding local maximum as the d/ℓ increases. More inter-
estingly, panels f , g, and h reveal that the longitudinal
drag first evolves from a broad zero into a local mini-
mum between tall peaks. By d/ℓ = 1.76 these peaks
have merged to form a single peak. Further increases of
d/ℓ steadily reduce the magnitude of this peak in Rxx,D.
Figure 2 displays both drag resistances precisely at
νT = 1 as functions of d/ℓ. These data were obtained
from sample B at T = 50mK. As the figure indicates, the
Hall drag resistance Rxy,D undergoes a rapid yet smooth
transition from very small values above d/ℓ ≈ 1.8 to the
very large value of h/e2 = 25.8kΩ for d/ℓ below about
1.65. At the same time, the longitudinal drag Rxx,D
exhibits a strong and rather symmetric peak in the tran-
sition region. At T = 50mK this peak is centered at
d/ℓ ≈ 1.73 and has a half-width of about ∆(d/ℓ) = 0.035.
The height of the peak, about 1.8kΩ, represents an im-
pressively large drag resistance. This value is in fact
roughly comparable to the conventional longitudinal re-
sistance Rxx of the bilayer system under the same con-
ditions (the deep minimum in Rxx characteristic of the
νT = 1 QHE developing only at lower values of d/ℓ).
Both the d/ℓ location and the height of this peak in
Rxx,D vary slightly from one sample to the next, but
its existence and qualitative behavior is quite robust.
The width and location of the peak in Rxx,D at νT = 1
3FIG. 2: a) Hall (open dots) and longitudinal (closed dots)
drag at νT = 1 and T = 50mK vs. d/ℓ in sample B. b) and c)
Temperature dependence of location and half-width of peak
in Rxx,D at νT = 1. Lines are guides to the eye.
depend upon temperature. Figure 2b reveals that the
peak moves to lower d/ℓ as T increases. This dependence,
which is roughly linear in T , extrapolates to about d/ℓ =
1.76 as T → 0. At the same time, Fig. 2c shows that the
peak half-width, ∆(d/ℓ), increases substantially as the
temperature is increased to 300mK. Interestingly, over
the range 25mK < T < 300mK the height of the peak in
Rxx,D at νT = 1 varies by only about 15%.
Figure 3 displays the temperature dependence ofRxx,D
at νT = 1 at three different values of d/ℓ. For d/ℓ = 1.58,
which is well within the bilayer QHE, Rxx,D rises as the
temperature is reduced from T = 0.5K to about 0.2K but
then drops precipitously as T is further reduced. In this
low temperature regime Rxx,D is thermally activated[7],
i.e. Rxx,D ∼ e
−EA/T , with EA ≈ 0.4K. This depen-
dence is expected within the gapped QHE phase and,
indeed, the conventional resistivity Rxx shows the same
activation energy[7]. At d/ℓ = 1.93, which is in the non-
QHE compressible phase, Rxx,D is much smaller in mag-
nitude and falls monotonically as T is reduced. This
temperature dependence is reminiscent of that seen at
B = 0 in the present samples[15] and at νT = 1 in much
more widely-spaced (d/ℓ ≈ 3.9) double layer 2D electron
systems[16]. Roughly speaking, this behavior reflects the
characteristic reduction of the phase space for inelastic
Coulomb scattering events as the temperature falls. A
quantitative model for Coulomb drag at νT = 1 and large
d/ℓ has been developed by Ussishkin and Stern[17].
At d/ℓ = 1.74, i.e. in the middle of the transition
region, the temperature dependence of Rxx,D at νT =
1 is markedly different than at both larger and smaller
FIG. 3: Temperature dependence of longitudinal drag resis-
tance at νT = 1 in sample B at three different d/ℓ values.
d/ℓ. After remaining roughly constant on cooling from
T = 0.5K to about 0.2K, the drag rises rapidly as the
temperature falls further. Below about T = 50mK Rxx,D
apparently saturates.
The data described above show that Coulomb drag is
a sensitive indicator of the transition between the weakly
and strongly-coupled regimes of a bilayer 2DES at νT =
1. This is especially clear from the behavior of the Hall
drag resistance Rxy,D shown in Fig. 2. Although non-
zero Hall drag can in principle result from density (or
energy) dependent scattering rates in a 2D system[18,
19], the development of a large, and ultimately quantized,
Rxy,D is generally believed to require non-perturbative
interlayer correlations. Indeed, the qualitative behavior
of Rxy,D shown in Fig. 2 was anticipated[20, 21, 22, 23,
24].
In contrast, the strong peak in the νT = 1 longitudi-
nal drag resistance Rxx,D which develops in the middle
of the Hall drag transition comes as a surprise. It seems
reasonable to interpret the d/ℓ value at the center of this
peak as the critical one separating the strongly-coupled
νT = 1 QHE phase from the weakly-coupled non-QHE
phase. Within the ferromagnetism picture of the QHE
phase, at zero temperature this critical point marks the
destruction of the ordered state by quantum fluctuations
of the pseudospin moment. The shifting of the peak to
lower d/ℓ as the temperature rises (cf. Fig. 2b) is con-
sistent with thermal fluctuations further destablizing the
ordered state. The non-zero width of the peak in Rxx,D
vs. d/ℓ suggests an inhomogeneous situation in which the
2D electron system fluctuates between the QHE and non-
QHE phases. Stern and Halperin (SH)[25] have suggested
that these fluctuations are static and result from meso-
scopic spatial inhomogeneities of the 2D electron density.
On the other hand, dynamic critical fluctuations in an
otherwise homogeneous system could also be involved.
In the SH picture, as d/ℓ is reduced toward the crit-
ical value puddles of the strongly-coupled QHE phase
appear within a background of weakly-coupled non-QHE
4FIG. 4: Longitudinal drag in sample C at T = 300mK. Ex-
panded view in the inset reveals a small enhancement near
νT = 1 where d/ℓ = 2.6.
fluid. As d/ℓ is reduced further, these puddles eventually
percolate. Via an analysis which assumes the puddles
are (as expected) counterflow superfluids while the back-
ground fluid is a conventional double layer 2D conductor
with a large Hall resistance but little Coulomb drag, SH
conclude that the macroscopically averaged longitudinal
drag resistivity ρxx,D of the composite system can be-
come very large just before the puddles percolate. In
this situation, SH predict that ρxx,D grows as the tem-
perature falls, eventually saturating near h/2e2 ≈ 13kΩ.
Figure 3 demonstrates that such a qualitative tempera-
ture dependence is observed. Although Rxx,D near the
midpoint of the transition region reaches only ≈ 1.9kΩ as
T → 0, classical geometric effects[26] associated with our
square sample geometry suggest that the experimental
longitudinal drag resistivity may be as much as a fac-
tor of π/ln2 larger than Rxx,D, or about 8.6kΩ. Further
experiments may reveal whether, as SH suggest, the re-
maining discrepancy is due to the finite conventional (i.e.
parallel transport) resistivity of the sample at νT = 1.
We turn finally to the behavior of Coulomb drag at
larger layer separations. Prior experiments[16], at d/ℓ ≈
3.9, showed no evidence of any anomaly in Rxx,D at to-
tal Landau level filling νT = 1. As Fig. 2 shows, the
enhancement of Rxx,D at νT = 1 in the present samples
subsides rapidly as d/ℓ increases. Surprisingly, however,
it remains observable out to d/ℓ ≈ 2.6. At these large
d/ℓ the enhancement appears as a small bump on top of
a background arising from drag scattering processes be-
tween weakly-coupled 2D layers - see Fig. 4. The bump
at νT = 1 is a genuine bilayer effect: It remains present
even when small antisymmetric density imbalances are
imposed on the double layer system. By contrast, the
other features (e.g. the minima at νT = 2/3 + 2/3 and
2/5+2/5) seen in Fig. 4 split in two, thus proving that
they are fundamentally single layer effects.
The existence of enhanced longitudinal drag at νT = 1
at such large d/ℓ is not understood. In principle, there
may be local regions in our samples in which d/ℓ has
been reduced by atomic steps in the various heterointer-
faces. However, the needed reduction is ∼ 30%, or about
8nm, and this seems implausibly large[27]. We emphasize
that no analogous anomaly is seen in the zero bias inter-
layer tunneling conductance. The enhanced tunneling,
which heralds the onset of interlayer phase coherence[6],
is either absent or unobservably small for d/ℓ ≥ 1.85.
This discrepancy raises the possibility that the enhanced
νT = 1 drag at larger d/ℓ may not require interlayer
phase coherence.
We thank Ady Stern, Bertrand Halperin and Steven
Girvin for enlightening discussions and Ian Spielman for
help with sample fabrication. This work was supported
by the NSF under Grant No. DMR0070890 and the DOE
under Grant No. DE- FG03-99ER45766.
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Double layer two-dimensional electron systems: Probing the transition from weak to strong coupling with Coulomb drag

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