Quasiparticle Tunneling across an Exciton Condensate
Ding Zhang ,
1,2,3
,*
Joseph Falson,
2
,§
Stefan Schmult,
2
,
†
Werner Dietsche,
2
and Jurgen H. Smet
2
,
‡
1
State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
2
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
3
Beijing Academy of Quantum Information Sciences, Beijing 100193, China
(Received 7 August 2019; revised manuscript received 2 January 2020; accepted 3 June 2020; published 19 June 2020)
The bulk properties of the bilayer quantum Hall state at total filling factor one have been intensively
studied in experiment. Correlation induced phenomena such as Josephson-like tunneling and zero Hall
resistance have been reported. In contrast, the edge of this bilayer state remains largely unexplored. Here,
we address this edge physics by realizing quasiparticle tunneling across a quantum point contact. The
tunneling manifests itself as a zero bias peak that grows with decreasing temperature. Its shape agrees
quantitatively with the formula for weak quasiparticle tunneling frequently deployed in the fractional
quantum Hall regime in single layer systems, consistent with theory. Interestingly, we extract a fractional
charge of only a few percent of the free electron charge, which may be a signature of the theoretically
predicted leakage between the chiral edge and the bulk mediated by gapless excitations.
DOI:
10.1103/PhysRevLett.124.246801
The
ν
tot
¼
1
=
2
þ
1
=
2
quantum Hall state (QHS)
[1]
is an
archetypical multicomponent state described by Halperin
’
s
(1,1,1) wave function
[2]
that keeps attracting attention
[3
–
8]
. It was first realized in GaAs double quantum well
systems
[9,10]
and more recently in graphene double
bilayers
[11
–
13]
. This ground state of the system can be
well understood as a condensate of excitons
[1,14]
. The
formation of such a condensate breaks the U(1) symmetry
of the system
[15]
, giving rise to gapless excitations, i.e.,
Goldstone modes in the bulk. In the pseudospin language,
such an excitation corresponds to the generation of a
quasiparticle referred to as a meron that can be viewed
as half a skyrmion
[16,17]
. A meron carries an electrical
charge of
ð
e=
2
Þ
but tends to bind a second meron with
opposite vorticity. Immediately following the early excite-
ment on the quantum Hall bilayer experiments
[18]
, Lopez
and Fradkin
[19]
constructed an effective field theory for
the edge states. They showed that the edge in a
ð
m; m; m
Þ
Halperin multicomponent QHS is largely analogous to that
of the
1
=m
Laughlin QHS itself (
m
is an integer), but there
exists leakage from the edge to the bulk due to the presence
of gapless excitations
[20]
[inset of Fig.
1(a)
]. It was
proposed that an interedge tunneling experiment should be
carried out to verify the theory and shed light on the edge-
bulk interactions
[19]
.
In vivid contrast to the extensive and fruitful exper-
imental studies of the bulk properties of quantum Hall
bilayer systems
[4,18,21
–
26]
, the edge physics has
remained unexplored for nearly two decades. While these
studies were for a long time the exclusive privilege of the
GaAs community, by now an abundance of multi-
component states has been unveiled in a number of
other material systems such as graphene based van der
Waals heterostructures as well as topological insulators
[6,11
–
13,27
–
30]
. Theoretical insights have suggested that
non-Abelian properties of excitations can be harnessed in
quantum Hall bilayers by manipulating its edge
[31
–
35]
.
All these developments make a disclosure of edge physics
in bilayers even more imperative. The need for both global
top and back gates in quantum Hall bilayers certainly
hampered progress, as there was little room for introducing
extra gates to manipulate the edge. With the advent of low-
density (
<
2
×
10
10
cm
−
2
) GaAs bilayer samples with high
quality (
μ
≥
1
×
10
6
cm
2
=
V s), it has become possible to
enter the
ν
tot
¼
1
state by only tuning a back gate
[36]
,
leaving the top surface available to deploy the split gate
technique for edge manipulation. The latter has been
widely used to address interedge tunneling for instance
in single layer electron systems at fractional filling factors
with both odd
[37]
and even
[38
–
42]
denominators. These
studies extracted effective charges that are overall close to
the theoretical expectations, although the extracted
Coulomb interaction strength may be influenced by the
device geometry
[43]
.
In this Letter, we successfully realize quasiparticle
tunneling in a GaAs bilayer and investigated its edge state
for the
ν
tot
¼
1
QHS. We obtain a clear zero bias peak in
the tunnel conductance of a quantum point contact (QPC).
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI. Open access publication funded by the Max Planck
Society.
PHYSICAL REVIEW LETTERS
124,
246801 (2020)
0031-9007
=
20
=
124(24)
=
246801(6)
246801-1
Published by the American Physical Society
The peak and its temperature dependence fit well to the
theoretical formula derived for weak quasiparticle tunnel-
ing in a single layer
[44]
, confirming the theoretically
predicted correspondence between the bilayer QHS and the
single layer states
[19]
. Interestingly, the extracted quasi-
particle charge is only a few percent of the free electron
charge, presumably reflecting a distinct type of interaction
between the quasiparticle on the edge and the excitonic
condensate in the bulk
[20]
. The presence of gapless
excitations in the bulk
[15]
allows for leakage from the
edge
[19]
and this process should be taken into account for
a complete theory of quasiparticle tunneling
[44]
in
bilayers.
The bilayers in this study reside in a double quantum well
of a GaAs
=
Al
x
Ga
1
−
x
As heterostructure grown by molecular
beam epitaxy
[36]
. Split gates made out of Ti/Au were
fabricated on top of the surface with the help of electron-
beam lithography, thermal evaporation and a lift-off pro-
cedure
[45]
. The sample was measured in two
3
He-
4
He
dilution refrigerators with a base temperature of about
20 mK. The total density
—
tuned by applying a back gate
voltage (
V
g
)
—
ranged from
4
.
5
–
6
.
0
×
10
10
cm
−
2
with the
balanced case corresponding to a density of
5
.
4
×
10
10
cm
−
2
.
The mobility exceeds
1
.
1
×
10
6
cm
2
=
V s. Resistances were
measured using heterodyne detection for an ac current
excitation between 0.2 and 0.5 nA at a frequency of 11.3
to 13.3 Hz.
Figure
1(b)
displays the resistance curves in the balanced
case by using the measurement configuration shown in
Fig.
1(a)
.
R
xx
and
R
xy
reflect the transport in the bulk
region. They show well-defined sequences of QHS for such
a balanced bilayer. In comparison,
R
D
and
R
−
D
are the
diagonal resistances measured across the QPC for positive
and negative magnetic fields, respectively.
R
D
and
j
R
−
D
j
significantly deviate from
R
xy
at filling factors below 2,
even if no voltage is applied to the split gate (
V
s
¼
0
). It
indicates that the QPC is already defined due to a work
function difference between the metallic gate material and
GaAs causing significant depletion of electrons in the
bilayer system. Measurements on a GaAs bilayer from
the same wafer with a global top gate confirm that the
metallic gate reduces the electron density by about 2 to
3
×
10
10
cm
−
2
and the regions beneath the metallic gates
with a density of about
2
×
10
10
cm
−
2
become well
insulating at magnetic fields exceeding 2 T. Current is
then forced to flow through the narrow constriction area
only and some current may get diverted to the other side of
the QPC. Even if so, the
R
D
ð
B
Þ
and
j
R
−
D
ð
−
B
Þj
curves still
exhibit the plateaus associated with the integer quantum
Hall effect at the same magnetic fields as those of the bulk
[see Fig.
1(b)
]. This overlap suggests that the density in the
QPC itself is close to the bulk value
—
a prerequisite for
weak quasiparticle tunneling. The deviation of
R
D
from
R
xy
at
ν
tot
<
2
reflects quasiparticle tunneling between the two
counterpropagating edges. According to the Landauer-
Büttiker formalism
[46]
, this interpretation would give rise
to an exact overlap between the calculated average (dotted
line) of the diagonal resistances,
ð
R
D
þj
R
−
D
jÞ
=
2
, and
R
xy
.
This is indeed the case, in particular also at the total filling
factor 1, i.e.,
ν
tot
¼
1
=
2
þ
1
=
2
.
The successful implementation of quasi-particle tunnel-
ing in a bilayer at
ν
tot
¼
1
can also be demonstrated by
studying the current and temperature dependence. When
applying a dc source-drain current (
I
dc
), a Hall potential
equal to
R
xy
I
dc
builds up between the two counterpropa-
gating edge states. It detunes them energetically and results
in a suppression of the quasiparticle tunneling. Since the
tunneling conductance is proportional to the diagonal
resistance, i.e.,
g
T
¼ð
R
D
−
R
xy
Þ
=R
2
xy
, a suppressed
g
T
at
finite
I
dc
manifests itself as a zero-bias peak in
R
D
.
Experimentally, we obtain such a zero-bias peak as shown
FIG. 1. (a) Schematic illustration of the quantum Hall bilayer
device and the measurement configuration. The upper-left inset
sketches the quasiparticle tunneling between two counterpropa-
gating edges of the
ν
tot
¼
1
state. (b) Hall, longitudinal, and
diagonal resistances as a function of magnetic field for the GaAs
bilayer in the balanced case measured at the base temperature of
the dilution refrigerator. The dotted curve is the calculated
average
ð
R
D
þj
R
−
D
jÞ
=
2
. (c) Diagonal resistance
R
D
as a function
of dc bias current at different magnetic fields (filling factors are
indicated). (d)
Δ
R
D
(see text) as a function of temperature. The
dotted line is a fit to the data points. The inset shows the
temperature dependence of the full width at half maximum
(FWHM) of the tunneling peak.
PHYSICAL REVIEW LETTERS
124,
246801 (2020)
246801-2
in Fig.
1(c)
. The peak feature depends sensitively on the
magnetic field and appears only around
ν
tot
¼
1
[45]
. The
dip at
ν
tot
¼
4
=
3
may stem from weaker interedge coupling
[37]
. Figure
1(d)
addresses the temperature dependence of
the peak at
ν
tot
¼
1
. The peak height
Δ
R
D
is defined as the
difference between the maximum value in each
R
D
−
I
dc
curve and the background resistance [illustrated in
Fig.
1(c)
]. According to theory, the peak height should
exhibit a power-law dependence
[44]
,
T
2
g
−
2
, where
g
measures the Coulomb interaction strength. We fit our data
between 30 and 65 mK and obtain a
T
−
1
.
5
0
.
1
dependence.
The inset of Fig.
1(d)
also demonstrates that the peak width
scales linearly with
T
and approaches zero at
T
¼
0
,as
expected from theory
[44]
and observed in single layer
systems
[38,47]
. As the temperature approaches base
temperature,
Δ
R
D
starts to saturate. The quasiparticle
tunneling possibly enters the strong tunneling regime as
the interedge coupling gets enhanced at lower temperatures
[40,47]
. However, given the very low electron density of our
sample (
<
3
×
10
10
cm
−
2
) and the relatively high contact
resistance (
∼
h=e
2
), difficulties in reducing the electron
temperature in step with the mixing chamber temperature
may also play an important role and be responsible for the
saturation. We therefore use the data in the range of 30 to
65 mK for the following quantitative analysis.
Since all features of the peak at
ν
tot
¼
1
seem to be
consistent with the scenario of weak quasiparticle tunnel-
ing, we employ the formula derived for single layer systems
in the weak-tunneling limit
[38,44]
to fit our data:
g
T
¼
AT
2
g
−
2
F
g;
e
I
DC
R
xy
k
B
T
þ
g
∞
;
F
ð
g;x
Þ¼
B
g
þ
i
x
2
π
;g
−
i
x
2
π
×
π
cosh
x
2
−
2
sinh
x
2
Im
Ψ
g
þ
i
x
2
π
:
ð
1
Þ
Here,
B
ð
x; y
Þ
and
Ψ
ð
x
Þ
are the Euler beta function and the
digamma function, respectively.
e
represents the fractional
charge of the quasiparticles.
g
is the Coulomb interaction
constant.
A
and
g
∞
(background conductance) are
constants.
Figure
2(a)
presents the fits to the experimental data
using Eq.
(1)
when the layers have equal densities. We fit
simultaneously the peaks (dotted lines) at ten temperature
points with
e
,
g
,
A
, and
g
∞
as the four fitting parameters.
The experimental results can be satisfactorily described by
the weak tunneling formula, as expected from the Lopez-
Fradkin theory
[19]
. The extracted interaction constant is
g
¼
0
.
30
. However, the fitted fractional charge
e
is only a
small fraction of the electron charge. The smallness of
e
can be further appreciated by evaluating the uncertainty of
the fitted values. Figure
2(b)
displays the relative error,
defined as the fit residual normalized by the experimental
noise, as has been done previously in single layer systems
[38,40,42]
. A fit error below 1 indicates that the corre-
sponding
g
and
e
values are allowed because the
fit is consistent with the data within experimental noise.
Figure
2(b)
shows that the region with a low fit error is
confined well below
e
¼
0
.
2
e
, much smaller than
0
.
5
e
for
a meron or
e
for a quasiparticle on the edge
[19]
.
We further study the quasiparticle tunneling, if an
imbalance exists between the density of the layers. In
transport, the
ν
tot
¼
1
state is robust against such an
imbalance
[36]
, but the nature of the underlying ground
states change. When ramping up
V
g
from 0.5 V to about
1.4 V, the QHS evolves continuously from the single layer
ν
t
¼
1
state to the
ν
tot
¼
ν
t
þ
ν
b
¼
1
=
2
þ
1
=
2
state and
finally into the
ν
tot
¼
1
=
3
þ
2
=
3
state. Here,
ν
t
and
ν
b
refer
to the filling factors of the top and bottom layer, respec-
tively. A larger
V
g
also gives rise to a higher density such
that the
ν
tot
¼
1
state shifts to a higher magnetic field. The
effective layer separation,
d=l
B
, therefore becomes larger
due to the decrease of the magnetic length
l
B
at a fixed layer
distance
d
, resulting in a weaker
ν
tot
¼
1
state
[36]
.
FIG. 2. (a) Tunnel conductance (solid curve) as a function of dc current for ten different temperatures. Also shown are fit curves
(dashed lines) generated with a single set of optimized fit parameters obtained by simultaneously fitting the formula for weak
quasiparticle tunneling for all traces and minimizing the total error. The fitted
g
and
e
are 0.30 and
0
.
036
e
. (b) Normalized fit error
[48]
as a function of
g
and
e
=e
. The circle marks the best fit point. The contours correspond to relative errors of 1, 1.1, and 1.2, as indicated
by the arrows.
PHYSICAL REVIEW LETTERS
124,
246801 (2020)
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Drastic changes occur in the tunnel conductance as
shown in Fig.
3
, even for rather small degrees of imbalance.
A deep minimum around zero bias is observed when
V
g
is
set to 0.9 and 0.95 V. The tunnel conductance suddenly
transforms into a peak at
V
g
¼
1
V. The background
conductance
g
∞
, which serves as a measure of the trans-
mission through the QPC, stays almost the same across this
gate voltage interval. This is very different from previous
experiments
[37,47]
showing dip-to-peak transitions. We
assign this observed behaviour to a crossover from the
breakdown of the quantum Hall effect to interedge tunnel-
ing. This is supported by the fact that for
V
g
¼
0
.
9
V,
g
T
is
almost zero at
I
dc
¼
0
, suggesting near perfect transmission
through the QPC. Although the split gate voltage
V
s
is left
unchanged for different
V
g
, the range of filling factor where
the incompressible
ν
tot
¼
1
state forms, shrinks as
V
g
increases from 0.9 to 1.1 V. This is manifested by a
narrowing of the quantum Hall plateau at
ν
tot
¼
1
[36,45]
. It effectively results in a reduction of the spatial
distance between the two edges. By going to higher
V
g
such that
ν
b
>
ν
t
, the peak height decreases because the
ν
tot
¼
1
state becomes weaker at a higher total density
[36]
.
In Fig.
4
we explore the influence of the small imbalance
among the two layers on the parameters
g
and
e
=e
required
to fit the quasiparticle tunnel conductance peak when
varying
V
g
near the balanced point between
V
g
¼
1
.
0
and 1.2 V. We consistently get a small charge of
0
.
1
e
or
even smaller. Similar results were obtained either from the
same QPC upon different thermal cycling or from another
QPC
[45]
. This reproducibility rules out the scenario of
incidental resonant tunneling with local charges in the QPC
as the mechanism for the reduced
e
[41]
. We also note that
the influence from the neighboring FQHS should be
minimal in the balanced situation as the individual layer
is at
ν
¼
1
=
2
. Figure
4
also includes previous experimental
results gathered on single layer systems at
ν
¼
5
=
2
[38,40
–
42]
and
ν
¼
1
=
3
[37]
with the same approach.
Analogously, for those works the extracted
g
spans a wide
range of values and exhibits a strong dependence on the
transmission through the QPC
[37,41,42]
, possibly via the
device geometry effect
[43]
. However, the extracted
e
=e
values in previous experiments all crowd around the
theoretically expected values (vertical dotted lines). The
variation in the Coulomb interaction due to different
confinement potential almost exclusively affects the esti-
mated
g
, whereas
e
apparently reflects an intrinsic
property of the quasiparticle. The small fractional charge
in our bilayer cannot be simply explained by the geometry
effect.
Instead, it may reveal the underlying interaction between
the edge and the bulk of the bilayer. We recall that Eq.
(1)
was derived for tunneling between chiral Luttinger liquids
across a gapped bulk
[44]
. In contrast, the
ν
tot
¼
1
state
possesses gapless excitations in the bulk
[15]
, which offers
an additional channel for the chiral edge to interact with.
This is referred to as an
“
open Luttinger liquid
”
[20]
. Lopez
and Fradkin have argued that this blurs the distinction
between the edge and the bulk. Excitations on the edge can
directly leak into the bulk. The same may occur in the
opposite direction on the other side and tunneling across the
bulk in the usual sense would not be required
[19]
. This
process is schematically depicted in Fig.
1(a)
and this
process would be nonlocal
[19]
. We speculate that the
FIG. 3. Tunnel conductance as a function of bias current at
different temperatures and for different degrees of imbalance
between the layers.
V
g
is varied from 0.9 V (top leftmost panel) to
1.25 V (bottom rightmost panel) in 0.05 V steps. The estimated
filling factor in the top (
ν
t
) and bottom (
ν
b
) layer have been
included in each panel. The split gate voltage to form the QPC is
kept constant at
V
s
¼
0
.
5
V after cycling to
−
1
.
5
V at base
temperature.
FIG. 4. Fitted values of
g
and
e
=e
for the
ν
tot
¼
1
state near the
balance point. Data points for
ν
¼
5
=
2
from Refs.
[38,40
–
42]
and
for
ν
¼
1
=
3
from Ref.
[37]
are also shown. The error bar for one
of the
ν
¼
5
=
2
data points stems from a variation of the
confinement while fixing
e
=e
¼
1
=
4
during fitting
[42]
. Empty
squares for
ν
¼
5
=
2
are extracted by varying the QPC gate
voltage
[41]
at a fixed magnetic field (we avoid taking the points
in the accidental resonant situation). Vertical dotted lines at
1
=
4
and
1
=
3
correspond to the expected ratio of
e
=e
for the
5
=
2
and
1
=
3
FQH states, respectively.
PHYSICAL REVIEW LETTERS
124,
246801 (2020)
246801-4
leakage effectively reduces the amount of charge that
reaches the other edge
[49]
.
In summary, interedge quasiparticle tunneling in a
quantum Hall bilayer was finally realized experimentally.
We observed the tunneling peak at the
ν
tot
¼
1
QHS. The
extracted fractional charge amounts to an unusually small
value
—
only a few percent of the electron charge, which
may be caused by the leakage from the edge to the bulk via
the gapless excitations.
We thank Klaus von Klitzing, Hailong Fu, Pengjie
Wang, Haiwen Liu, Xincheng Xie, and Timo Hyart for
fruitful discussions. We thank Marion Hagel for technical
assistance. Self-consistent simulations of the quantum point
contact were carried out with the nextnano software
package. This study was financially supported by the
National Natural Science Foundation of China (Grants
No. 11922409, No. 11790311, and No. 11604176) and the
German Ministry of Education and Research (BMBF Grant
No. 01BM900).
*
dingzhang@mail.tsinghua.edu.cn
†
Present address: TU Dresden, Electrical and Computer
Engineering, Institute of Semiconductors and Microsys-
tems, Nöthnitzer Str. 64, 011187 Dresden, Germany.
‡
j.smet@fkf.mpg.de
§
Present address: Department of Applied Physics and
Materials Science, California Institute of Technology,
Pasadena, CA, USA.
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[45] See the Supplemental Material at
http://link.aps.org/
supplemental/10.1103/PhysRevLett.124.246801
for (1)
the simulation of the potential profile and density in the
region of the quantum point contact, (2) key transport
quantities derived from the Landauer-Büttiker formalism,
and (3) additional data sets of the tunnel conductance near
ν
tot
¼
1
for two different QPCs.
[46] Based on Landauer-Büttiker formalism, the longitudinal
resistance across the QPC
R
T
follows:
R
T
¼
R
D
−
R
xy
.We
show in the Supplemental Material that this is indeed the
case. Furthermore,
R
T
ð
I
dc
Þ
gives rise to essentially the same
result as
R
D
ð
I
dc
Þ
.
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absolute values of fit residuals at a fixed combination of
g
and
e
=e
. This value is further divided by
0
.
004
h=e
2
—
the
noise of measurement. This measurement noise is evaluated
from the resistance fluctuation in the high bias range
(
I
dc
from
−
14
to
−
11
nA): we take three times the standard
deviation
σ
in this range.
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