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Supplementary information: quasi-particle tunneling across an
exciton condensate
Ding Zhang
1
;
2
, 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
(Dated: May 6, 2020)
Abstract
In the supplementary information, we present simulation results of the potential profile and
density in the region of the quantum point contact (QPC). We also discuss the key transport
quantities derived from the Landauer-B ̈uttiker formalism and include additional data sets of the
tunnel conductance near

tot
= 1 for two different QPCs.
1
SIMULATION OF THE QUANTUM POINT CONTACT
Figure S1 shows the Hall bar structure together with the split gates that define the
quantum point contact (QPC). The split gates have a nominal separation of 4

m. They
are vertically displaced from the quantum wells by a spacer of approximately 710 nm. As
a result of this large separation, the orifice of the quantum point contact potential of the
two-dimensional electron gas (2DEG) is much narrower. To simulate this narrowing, 3D
self-consistent Poisson calculations were carried out with the nextnano software (nextnano
GmbH). Figure S2 displays the modeled structure and the calculated results. The real
wafer contains in-situ grown LT-GaAs and multiple superlattices on the substrate. This
complex heterostructure is simplified to the symmetric geometry shown in Fig. S2. To
minimize the computation time, the doping levels used were obtained by first running 1D
self-consistent Poisson-Schr ̈odinger tests until the density values in the two quantum wells
match experimental values in the balanced case [Fig. S2(a)]. In the following simulation, we
study the influence of top gates on this heterostructure. A gate voltage of
0
:
5 V is applied
to the two top gates to simulate the depletion effect brought about by the work function
difference between the metallic gates and the semiconductor.
Close to the center (
x
= 0
; y
= 0) of the QPC, the split gates do not affect the 2DEG.
Hence, the conduction band profile along the
z
-direction at
x; y
= 0 reflects the band
structure in the bulk. Fig. S2 (c) shows 2D colored plots of the density distribution. The
top and bottom panels are slices at the top and bottom quantum well, while the middle
panel illustrates the vertical density distribution. From these plots it is apparent that the
QPC opening in the top layer is narrower than in the bottom layer.
In order to assess the effective width of the QPC, the percentage density change in the
upper and lower quantum well is plotted in panel (d) of Fig. S2. Starting from the center
of the QPC and moving to the region beneath the gate, the density drops down to zero
and the relative density change approaches 100%. Although full depletion is only reached
underneath the gate, a wide region not covered by the gate is affected. In the balanced
case the

tot
= 1 quantum Hall state only tolerates a small density variation. Fig. S3 plots
the longitudinal resistance in the vicinity of the

tot
= 1 state for different total densities,
i.e. different degrees of layer-imbalance. At higher backgate voltage, the total density is
increased and the

tot
= 1 state occurs at higher magnetic field. This shift to higher fields is
2
accompanied by a reduction of the width of the longitudinal resistance minimum signaling
the

tot
= 1 state. We estimate this width (∆
B
) from the distance between the vertical
dotted bars. The density tolerance, defined as ∆
n=n
tot
, is proportional to ∆
B=B
0
(
B
0
is
the central magnetic field of the

tot
= 1 state). Figure S3(b) shows that ∆
n=n
tot
decreases
with increasing total density. For the balanced case at a total density of 5
:
44

10
10
cm
2
,
the

tot
= 1 state only tolerates a relative density change of up to 1.5%. The resistance
trace corresponding to the balanced case is also plotted as an inset to panel (d) of Fig. S2.
The dotted vertical line in Fig. S2(d) marks the region within the QPC where the

tot
= 1
survives. Regions with a relative density change larger than 1.5% become insulating at this
magnetic field. The opening of the QPC in the 2DEG is therefore effectively about 2.4

m
in the balanced regime. More generally, when applying higher backgate voltages the QPC
opening narrows.
LANDAUER-B

UTTIKER FORMALISM
Applying the Landauer-B ̈uttiker formalism to our measurement configuration yields the
following equations:
R
D
=
h
e
2
1

QPC
;
(1)
R
D
=
h
e
2
(
2

bulk
1

QPC
)
;
(2)
R
T
=
h
e
2
(
1

QPC
1

bulk
)
:
(3)
R
D
(
R
D
) refers to the diagonal resistance value for
B >
0 (
B <
0).
R
T
is the longitudinal
resistance across the QPC. The difference between
R
D
and
j
R
D
j
reflects the chirality of the
edge current. This can be further verified by comparing (
R
D
+
j
R
D
j
)
=
2 with
R
xy
. From the
equations, we find:
R
D
+
j
R
D
j
2
=
h

bulk
e
2
:
(4)
As shown in the main text, although
R
D
and
R
D
are measured at opposite magnetic fields,
(
R
D
+
j
R
D
j
)
=
2 simply collapses back to
R
xy
in the quantum Hall regime. This overlapping
behavior supports the existence of edge channels in this quantum Hall bilayer system.
3
In Fig. S4, we also show data of the same sample, but measured in a different refrigerator.
There, not only
R
D
,
R
D
, but also
R
T
was measured. This was one of the early measurements
with temperature and noise conditions that were not optimized. The data presented in the
main text as well as in Fig. S5 and S6 were measured at reduced electron temperature, noise
level as well as lower excitation current. Despite the unoptimized conditions, the data in
Fig. S4 demonstrate that the results are in agreement with Landauer-B ̈uttiker formalism.
Combining Eqs. (1) to (3), one obtains:
R
D
j
R
D
j
2
=
R
T
:
(5)
The lower panel of Fig. S4(a) confirms the relation above by comparing (
R
D
j
R
D
j
)
=
2 with
R
T
R
xx
. In the quantum Hall regime,
R
xx
= 0 such that (
R
D
+
R
D
)
=
2 and
R
T
R
xx
overlap. Comparing Eq. (1) and Eq. (3), one sees that:
R
T
=
R
D
R
xy
:
(6)
This is also confirmed in Fig. S4(a).
Fig. S4(b) also shows that
R
T
(
I
dc
) and
R
D
(
I
dc
) both exhibit the zero bias peak. They
share the same peak height and width, which are essential in the fit for determining both
e

and
g
. We illustrate this by vertically shifting the curve fitted to
R
D
(
I
dc
) to overlap with
the data of
R
T
(
I
dc
).
EXTENDED DATA
We show in Fig. S5 and S6 the fitting and error analysis of tunneling peaks for different
degrees of imbalance between the top and bottom layer recorded during a second cool-down
and for a second quantum point contact.
4
FIG. S1. Optical images of the patterned GaAs bilayer. The inset illustrates the design of the
quantum point contact. Indicated lengths are in

m.
5
FIG. S2. Simulation of the quantum point contact (QPC) in the bilayer. (a) Conduction band
profile at the center of the QPC (
x
= 0
; y
= 0). (b) GaAs double quantum well heterostructure used
in the 3D simulation. Dimensions are in nm. (c) 2D cuts of the calculated 3D density distribution.
The top and bottom color plot display the in-plane density variation inside the top and bottom
quantum well, respectively. The panel in the middle shows the density distribution in the vertical
direction. (d) Density variation from the center
y
= 0 to the edge
y
= 3200 nm. The plotted
quantity is the relative density change. The density at the center of the QPC (
y
= 0) serves as
the reference value. Red and blue squares represent the spatial dependence in the top and bottom
quantum well. Black squares correspond the spatial dependence of the total density. The inset
illustrates the magnetic field range over which the

tot
= 1 quantum Hall state is observed for the
balanced case.
6
FIG. S3. (a) Longitudinal resistance measured around the

tot
= 1 state for different values of the
total density. The curves have been recorded for increasing back gate voltages from 0.9 V (bottom
trace) to 1.25 V (trace at the top) with a voltage step of 0.05 V. They are vertically offset for
clarity. The number near each curve is the total density in units of

10
10
cm
2
. Dotted vertical
bars demarcate the quantum Hall plateau/minimum. The distance between the vertical bars is
referred to as ∆
B
and serves as a measure for the density change that can be tolerated before the

tot
= 1 state disappears, ∆
n=n
tot
= ∆
B=B
0
. (b) ∆
n=n
tot
as a function of the total density.
7
FIG. S4. (a) Magneto-resistance and Hall resistance traces. The inset explains the different
quantities measured in the field sweep.
R
D
and
R
D
were obtained by sweeping
B
from 1.5 to 2.5
T and -1.5 to -2.5 T, respectively. (b) Current bias dependence of the longitudinal resistance (
R
T
)
across the QPC as well as the diagonal resistance (
R
D
). Here
R
D
is subtracted by a constant of
25.813 kΩ to enable comparison. The solid red line is a fit to the
R
D
(
I
dc
) data. The electron
temperature is assumed equal to 40 mK instead of the base temperature. This follows from the
observation that the tunneling peak saturates at
T

40 mK in this dilution refrigerator. The
dashed line is the same fit but shifted downward.
8