Present address: ]Google Quantum AI, Mountain View, CA, USA

# Evidence for chiral supercurrent in quantum Hall Josephson junctions

Hadrien Vignaud  These authors contributed equally to this work.  Univ.
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
David Perconte  These authors contributed equally to this work.  Univ.
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Wenmin Yang  Bilal Kousar  Edouard Wagner  Frédéric Gay  Univ. Grenoble Alpes,
CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France  Kenji Watanabe
Research Center for Electronic and Optical Materials, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi Taniguchi
Research Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Hervé Courtois  Univ.
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Zheng Han  State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P. R. China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan
030006, P. R. China  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel,
38000 Grenoble, France  Hermann Sellier  Univ. Grenoble Alpes, CNRS, Grenoble
INP, Institut Néel, 38000 Grenoble, France  Benjamin Sacépé
<EMAIL_ADDRESS>[  Univ. Grenoble Alpes, CNRS, Grenoble INP,
Institut Néel, 38000 Grenoble, France

###### Abstract

Hybridizing superconductivity with the quantum Hall (QH) effects has major
potential for designing novel circuits capable of inducing and manipulating
non-Abelian states for topological quantum computation[1, 2, 3]. However,
despite recent experimental progress towards this hybridization[4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16], concrete evidence for a chiral QH Josephson
junction[17, 18] –the elemental building block for coherent superconducting-QH
circuits– is still lacking. Its expected signature is an unusual chiral
supercurrent flowing in QH edge channels, which oscillates with a specific
$2\phi_{0}$ magnetic flux periodicity[17, 18, 19, 20, 21] ($\phi_{0}=h/2e$ is
the superconducting flux quantum, $h$ the Planck constant and $e$ the electron
charge). Here, we show that ultra-narrow Josephson junctions defined in
encapsulated graphene nanoribbons exhibit such a chiral supercurrent, visible
up to 8 teslas, and carried by the spin-degenerate edge channel of the QH
plateau of resistance $h/2e^{2}\simeq 12.9$ k$\Omega$. We observe reproducible
$2\phi_{0}$-periodic oscillation of the supercurrent, which emerges at
constant filling factor when the area of the loop formed by the QH edge
channel is constant, within a magnetic-length correction that we resolve in
the data. Furthermore, by varying the junction geometry, we show that reducing
the superconductor/normal interface length is pivotal to obtain a measurable
supercurrent on QH plateaus, in agreement with theories predicting dephasing
along the superconducting interface[21, 22, 23, 24]. Our findings mark a
critical milestone along the path to explore correlated and fractional QH-
based superconducting devices that should host non-Abelian Majorana and
parafermion zero modes[25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

The correlated and topological orders in the multifaceted QH effects have
significant potential as a resource for inducing topological superconductivity
in superconducting QH hybrid circuits[25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35]. Several blueprints for the experimental realization of Majorana zero
modes or their fractionalized generalization, the parafermions and Fibonacci
anyons, have been drawn up on the basis of hybridizing spin-polarized[26, 32,
33, 36] or fractional QH states[26, 2, 3]. Owing to their non-local nature and
non-commutative braiding properties, these non-Abelian zero modes are expected
to be the basis for fault-tolerant topological quantum computation[37, 1, 30,
3].

However, as appealing as this approach may be, coupling superconductivity and
QH effect remains an outstanding experimental challenge. The main dilemma
arises from the perpendicular magnetic field, $B$, required for the QH effect
in a two-dimensional electron gas (2DEG). While magnetic field generates and
strengthens the QH effect by increasing energy gaps in the Landau level
spectrum, it adversely affects superconductivity until breakdown at the upper
critical field. Furthermore, QH effect profoundly modifies the charge transfer
process at the superconductor/QH interface –the Andreev process converting an
incident electron into a retro-reflected hole. There, electrons and Andreev-
reflected holes have the same chirality, that is, co-propagating forward and
forming chiral Andreev edge states (CAES) along the interface[38, 39]. In a
Josephson junction connecting two superconducting electrodes through a 2DEG in
the QH regime, the CAES and QH edge states form a chiral loop that connects
both electrodes (see Fig. 1a). This yields a chiral supercurrent that is non-
locally split between the two edges of the 2DEG, and subjected to Aharonov-
Bohm quantum interference with a $2\phi_{0}$ flux periodicity, twice that of
conventional Josephson junctions[17, 18, 19, 20, 21].

Figure 1: Josephson effect through quantum Hall edge channels. a, Schematics
of the Josephson junction consisting of a graphene nanoribbon encapsulated in
hexagonal boron nitride (hBN) resting atop a graphite back-gate electrode.
Edge contacts are made of superconducting MoGe electrodes. Under magnetic
field, $B$, charge carriers of the QH edge channel undergo successive Andreev
reflections along the superconducting electrode, which convert incident quasi-
electrons (blue channel) into quasi-holes (red channel) and vice-versa, as
illustrated with semi-classical cyclotron trajectories in the insert. This
electron-hole mixture forms a CAES (dashed blue-red channel) along the
interface. b, Differential resistance $\text{d}V/\text{d}I$ of device HV88-B
measured at 8 T with a dc current bias of 1.2 $\upmu\rm{A}$ and plotted as a
function of back-gate voltage, showing a well quantized resistance plateau at
$\nu=2$. c, $\text{d}V/\text{d}I$ as a function of back-gate voltage and dc
current bias over the plateau region (grey area in b). Superconducting pockets
(in black) alternate with finite resistance regions. d, Superconducting $I-V$
curves at $V_{\rm g}=0.41$ V (white line in c). The hysteresis is
characterized by separate switching and retrapping currents. e, Shapiro map of
differential resistance measured in device HV88-C as a function of RF power
($f=1$ GHz) and dc current bias at $\nu=2.15$ and $B=3.45$ T.

Experimentally, employing graphene in contact with high upper critical field
superconductors has led to recent advances in superconducting-QH hybrid
devices[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], with evidence of CAES
along superconducting interfaces[12, 15] and crossed Andreev conversion
through narrow contacts[9, 14]. In 2016, a supercurrent in graphene Josephson
junctions in the QH regime was reported[8], visible at relatively low magnetic
field ($<2$T) and high filling factors ($\geq 6$). Yet, the junction exhibited
a standard $\phi_{0}$-periodic oscillation, in contradiction with the chiral
supercurrent prediction. Further work[11] suggested that additional conduction
channels induced by charge accumulation along etched graphene edges may yield
two trivial Josephson junctions in parallel, one per edge, explaining the
SQUID-like $\phi_{0}$-oscillation.

Here, we report on the observation of a chiral supercurrent carried by a
single, spin-degenerate QH edge channel in a graphene QH-Josephson junction at
bulk filling factor $\nu=2$. Our approach builds on theoretical predictions
pointing out that the supercurrent amplitude is inversely proportional to the
2DEG perimeter[17, 18, 19, 20], and that long superconductor-QH interfaces are
detrimental to coherence[21, 22, 23, 24]. We therefore purposely designed
ultra-narrow graphene Josephson junctions, with contact width of 125 to 330 nm
(see Extended Data Table 1), an order of magnitude smaller than any previous
work[4, 6, 8, 11], which allows us to unveil the chiral supercurrent.

We fabricated QH-Josephson junctions with exfoliated graphene nanoribbons
encapsulated between hexagonal boron-nitride flakes[40]. The resulting
heterostructures lay atop exfoliated graphite flakes that serve as back-gate
electrode to tune the charge carrier density with a voltage $V_{\rm g}$ (Fig.
1a). We chose MoGe as superconducting contact material, with a high upper
critical field ($H_{c2}\simeq 12.5$ T, see SI) and a contact resistance of
$460\pm 21$ $\Omega.\upmu$m averaged from 13 devices. To ensure a good
quantization of the QH plateau[41, 42], we designed junctions with aspect
ratio $L/W\sim 1$ , where $W$ is the width of the superconductor-graphene
contact and $L$ the distance between the two contacts (see Extended Data Table
1). All measurements are performed at a temperature of $0.01$ K.

Supercurrent on $\nu=2$ quantum Hall plateau

An essential benchmark for demonstrating superconducting transport through QH
edge channels is the conjunction of a well-defined supercurrent and a QH
plateau in the normal state. This is validated in Fig. 1b and c that display
the two-terminal resistance as a function of back-gate voltage at $8$ T for a
junction with $L=140$ nm and $W=180$ nm. Figure 1b displays the normal state
resistance measured at a high current bias of $1.2\,\upmu$A, which shows a
well quantized $h/2e^{2}$ plateau, indicating edge transport without bulk
backscattering. Concomitantly, the differential resistance map at low current
bias in Fig. 1c reveals pockets of supercurrent with zero resistance state
color coded in black. Individual current-voltage characteristics (Fig. 1d)
show a well-defined, hysteretic supercurrent with zero voltage drop below a
switching current of $0.56$ nA.

To get insight into the current-phase relation of the junction, we performed
Shapiro steps measurements under irradiation by a microwave tone of frequency
$f=1$ GHz. Fig. 1e displays a typical map of $\text{d}V/\text{d}I$ as a
function of current bias and microwave power, measured on the $h/2e^{2}$ QH
plateau. As the microwave power is ramped up, Shapiro voltage steps develop
and are visible as zero $\text{d}V/\text{d}I$ areas color coded in black. The
height of these steps is $2.0\,\upmu\rm{V}$ (see Extended Data Fig. 6) in
agreement with the theoretical expectation $hf/(2e)=2.1\,\upmu\rm{V}$. This
observation was reproduced at different frequencies (see SI), and indicates a
conventional $2\pi$-periodic current-phase relation, as expected for the spin-
degenerate QH edge channel at filling factor 2 (ref.[17, 19, 20]).

Quantum interference of chiral supercurrent

We now turn to the central result of this work, that is, demonstration of the
distinct $2\phi_{0}$ flux periodicity of the chiral supercurrent. Supercurrent
oscillations are readily seen in Fig. 2a and b, which display two typical
$\text{d}V/\text{d}I$ maps as a function of magnetic field and current bias,
at filling factor $\nu=1.9$ and $2.2$, respectively (see SI for the evaluation
of $\nu$). At constant filling factor (non-constant back-gate voltage),
superconducting pockets are observed over a large $B$-range showing periodic
oscillations. To evaluate the periodicity of the supercurrent oscillations, we
superpose on Fig. 2a and b a comb of white dashed lines equally spaced by
$2\phi_{0}/A$, that is, twice the superconducting flux quantum divided by the
effective graphene area. Clearly, maxima of the supercurrent oscillation match
the $2\phi_{0}$ flux periodicity, the hallmark of the chiral supercurrent.
Slight shifts of the oscillation are sometimes seen, such as for the last two
white dashed lines above 7 T in Fig. 2b.

Figure 2: $2\phi_{0}$-periodic chiral supercurrent oscillations. a, b,
Differential resistance $\text{d}V/\text{d}I$ at constant filling factors
$\nu=2.2$ in (a) and $\nu=1.9$ in (b) as a function of magnetic field and dc
current bias in device HV88-B. The two groups of white lines indicate the
expected period for $2\phi_{0}$-periodic oscillations involving 70% and 80% of
the graphene area on the left and right combs, respectively, in a and 70% of
the graphene area in b. c, d, Maps of the switching current and of the zero
bias differential resistance as a function of magnetic field and filling
factor, respectively. f, g, Fourier transforms of the magnetic field dependent
oscillations of the supercurrent map in (c) and the differential resistance
taken at a finite bias of 4 nA from the set of $I-V$ curves used in (d),
respectively, as a function of frequency $1/\Delta B$, equivalent area $A$ for
$2\phi_{0}$-periodic oscillations, and filling factor. Magnetic field ranges
used are 2T to 7.5T in (f), and 4T to 10T in (g). For both Fourier transforms,
a slowly varying background has been subtracted. The white solid lines
indicate the frequency corresponding to the $2\phi_{0}$-periodic oscillations
for the geometric area, whereas the white dashed lines take into account a
correction of the area $(L-2l_{B})\times(W-2l_{B})$ that accounts for a better
estimate of the positioning of the QH edge states at $l_{B}$ (taken at the
average value of 4.5T in (f) and 7 T in (g)) from the physical edge[43]. The
white error bars indicate the experimental uncertainty on the geometry
evaluation (see Methods and Extended Data Fig. 5). e, Schematics of the Landau
levels and of the position of their edge states in the device for two
different filling factors at a fixed magnetic field. h, Fourier transforms of
the differential resistance versus magnetic field at constant filling factor
for junctions with different sizes (devices B to F from bottom to top, sample
HV88). The color-coded rectangles indicate the frequencies for $2\phi_{0}$
oscillation based on the geometric area. Their width corresponds to the
uncertainty in the geometric area.

The key to this finding lies in analyzing the supercurrent oscillation at
constant filling factor instead of fixed back-gate voltage. Indeed, for the
latter, the charge carrier density is constant and the filling factor
decreases with $B$, leading to an inward displacement of the QH edge channels
relative to the physical edge of the graphene flake and hence a decrease in
the effective area. For our ultra-narrow junctions, such an area variation is
sufficient to alter the flux periodicity (see SI Fig. S9). Conversely, at
constant filling factor, that is, at constant Fermi level position in the
cyclotron gap, the edge channel remains at a nearly $B$-independent spatial
position, enabling us to unveil the $2\phi_{0}$ flux periodicity.

To substantiate our findings we present in Fig. 2c a full mapping of the
supercurrent (see Methods) over the $h/2e^{2}$ QH plateau. The
$2\phi_{0}$-periodic supercurrent oscillations form fringes with negative
slope, extending all over the plateau, on top of a slowly varying supercurrent
background. Figure 2f shows a Fourier transform of these fringes performed at
constant filling factor. The peak at $1/\Delta B$, which indicates the period
$\Delta B$ of the supercurrent oscillations, is close but slightly smaller
than the value expected for $2\phi_{0}$-periodic oscillations related to the
geometric graphene area (see white solid line and error bar related to the
uncertainty in the nanoribbon width). This slightly smaller effective area can
be accounted for by taking into account the distance between the actual edge
state and the physical graphene edge, which is of the order of the magnetic
length $l_{\rm B}=\sqrt{\hbar/eB}$ (see white dashed line). The close match
between the periodicity and the corresponding geometric graphene area
indicates that the QH edge channel is located in the immediate vicinity of the
crystal edge, in agreement with recent scanning tunneling spectroscopy
results[43].

In addition, one can notice a shift of the peak to higher frequency upon
increasing $\nu$. This shift reflects the increase in area with $\nu$ due to
the slight displacement of the QH edge channel towards the graphene edge, as
sketched in Fig. 2e, therefore providing a signature of the Landau level
dispersion at the edge. From Fig. 2f, we obtain an area variation of
$4.5\times 10^{-3}$ $\upmu$m2, which corresponds to an edge channel
displacement of $7$ nm, i.e. of the order of the magnetic length.

Importantly, the supercurrent oscillations are accompanied by resistance
oscillations of the resistive state, as seen in Figs. 1c, 2a and 2b. The
$\text{d}V/\text{d}I$ map at zero bias in Fig. 2d shows similar fringes with
negative slope as for the critical current in Fig. 2c (both maps are extracted
from the same set of data). The Fourier transform of the resistance
oscillations (Fig. 2g) also reveals a $2\phi_{0}$ flux periodicity with the
same frequency shift with $\nu$ as the supercurrent. We ascribe these
resistive state oscillations to a consequence of the Aharonov-Bohm phase
picked by the charge carriers circulating around the QH channel loop, while
undergoing chiral Andreev reflections along the superconducting interfaces.
Note that, contrary to usual Josephson junctions, maxima of supercurrent do
not necessarily coincide with minima of resistive state (see e.g. Fig. 2a and
b). This is not surprising as the phase accumulated by CAES differs from that
accumulated in the normal state by an additional phase shift picked up upon
Andreev reflections.

Inspecting Figs. 2c and d more closely, we see a decrease in the oscillation
period with magnetic field, which translates to an increase in effective area.
This variation can be accounted for by a decrease in the width of the Landau
level wavefunction that scales as $l_{\rm B}$, which in turn results in the
displacement of the QH edge channels towards the graphene edge (see Fig. 3b).
Here, unlike Figs. 2f, g and e, the area variation is not related to the edge
dispersion of the Landau level, but to the wavefunction shrinkage. An
approximate correction of the flux area by the magnetic length
$(L-2l_{B})\times(W-2l_{B})$, assuming a displacement of $l_{B}$ relative to
the graphene edges, is shown as a grey dashed line in Fig. 3a. This correction
is superimposed in Fig. 3a on the Fourier transform of Fig. 2d, which is
computed here in a $B$-window sliding along the $B$-axis at fixed filling
factor, and fits remarkably well the observed shift. Such a fine analysis of
the area variation due to the wavefunction shrinkage with magnetic field
confirms that the $2\phi_{0}$ flux periodicity is observed over the whole
range of magnetic field.

Figure 3: Edge state wavefunction shrinkage. a, Fourier transform of the
differential resistance at filling factor $\nu=1.9$ and finite current bias of
3 nA of the same data as in Fig. 2d, computed in a magnetic field window of
$3$ T sliding along the magnetic field axis, and plotted as a function of
inverse period $1/\Delta B$ and magnetic field. It reflects the $B$-dependence
of the oscillation period visible in Fig. 2d. A slowly varying background has
been subtracted. The white solid line indicates the frequency corresponding to
the $2\phi_{0}$-periodic oscillations for the geometric area $W\times L$. The
dashed line shows the expected frequency with a correction of the area as
$(L-2l_{B})\times(W-2l_{B})$. b, Schematics of the edge state position at the
edge of the graphene flake for two different values of magnetic field. The
width of the gaussian wavefunction of the zeroth Landau level is proportional
to the magnetic length $l_{B}$, which decreases with $B$. As a result, the
mean position of the edge channel shifts towards the physical edge upon
increasing $B$, leading to an increase of the effective area. The area
variations set by the Landau level wavefunction shrinkage illustrated in (b)
are in excellent agreement with the observed frequency shift in (a).

We obtained equivalent results for six other junctions for both critical
current and resistive state oscillations (see SI). As a summary, we present in
Fig. 2h the Fourier transforms of the oscillation for five junctions of
different sizes, which show an excellent agreement with the expected
$2\phi_{0}$ flux periodicity indicated by colored rectangles. Such a
systematic set of data therefore provides a conclusive demonstration of the
$2\phi_{0}$ flux periodicity of the chiral supercurrent.

Velocity renormalization at superconducting interface

Figure 4: Aharonov-Bohm interferometry with chiral Andreev edge states. a,
Checkerboard pattern of the differential resistance oscillation (after
background subtraction) as a function of the measured voltage and magnetic
field, at a fixed filling factor for device HV88-B. b, Fourier transform
amplitude of (a) as a function of measured voltage, showing a lobe structure.
The solid line is a fit using the theory of QH Fabry-Pérot interferometers of
Ref.[44] with a bias asymmetry parameter $x=0.02$. The maxima of the two side
lobes are separated by the Thouless energy, which is plotted in (c) for
various devices as a function of the inverse of the superconducting interface
width $1/W$. Different points for a given sample correspond to measurements at
different filling factors. The black line is a linear fit with the slope
yielding a renormalized edge velocity of $2.1\pm 0.5\times 10^{4}$ m/s. d,
Differential resistance map of device HV88-C at $3$ T as a function of filling
factor and measured dc voltage across the junction. The QH breakdown at the
edge of the plateau is visible at high voltage, as indicated by the two black
arrows. e, Linecuts measured at the same filling factor and magnetic field as
in (d) but with higher resolution and lower bias, showing superconducting
pockets as those seen in Fig. 2c. The oscillations disappear on a scale of the
order of the Thouless energy, while the large-scale fluctuations subsist up to
about 1.4 m$\rm{V}$. Note that the $2\phi_{0}$-periodic oscillations are
visible at low voltage bias as indicated by the dashed black lines.

We now analyse the energy dependence of the resistance oscillations. Figure 4a
shows $\text{d}V/\text{d}I$ as a function of the measured voltage across the
junction and magnetic field at constant filling factor $\nu=2.6$. The
$2\phi_{0}$-periodic oscillations undergo a phase shift of $\pi$ at finite
bias $V=155\,\upmu$V, forming a checkerboard pattern resembling that of Fabry-
Pérot QH interferometers[44, 45]. While latter contain a loop of QH edge
channels partially transmitted through quantum point contacts, here the
configuration is different since the quantum point contacts are replaced by
superconducting interfaces, which are partially reflecting the QH edge states
through Andreev processes with finite transparency. The checkerboard pattern
thus results from the phase shift $4\pi\delta\epsilon/E_{\rm Th}$ due to the
finite energy $\delta\epsilon$ of the injected electrons, where $E_{\rm
Th}=h/\tau$ is the ballistic Thouless energy related to the traveling time
$\tau$ along half of the loop perimeter. Following Ref.[44], we plot the
Fourier transform of the checkerboard pattern in Fig. 4b and extract $E_{\rm
Th}=640\,\upmu$eV. Importantly, this value is smaller than what is expected
for a QH edge state velocity $v_{\rm QH}=1.4\times 10^{5}$ m$/$s measured in
QH interferometers [44], which would give $E_{\rm Th}=1.8$ meV for the loop
with half-perimeter $L+W=318$ nm. This estimate is instead consistent with the
data at 14T above the critical field of the electrode shown in Extended Data
Figure 5, in which we measured $E_{\rm Th}=1.6$ meV.

This discrepancy can be understood if we consider the CAES velocity
renormalization along the superconducting interface[39, 21]. Due to the time
delay of the Andreev process, that is, the extra time of the order of
$h/\Delta$ that an impinging electron spends in the electrode with a
superconducting gap $\Delta$, the edge velocity $v_{\rm SC}$ along the
superconducting interface drops significantly. As a result, $\tau=(L/v_{\rm
QH}+W/v_{\rm SC})$ and the Thouless energy is dominated by the interface width
$W$, $E_{\rm Th}\sim hv_{\rm SC}/W$, when $v_{\rm SC}\ll v_{\rm QH}$. From the
analysis of four junctions of different sizes (see other checkerboards in
Extended Data Fig. 7), taken between 4.5 and 6 T, the evolution of $E_{\rm
Th}$ with $1/W$ shown in Fig. 3c gives an estimate of $v_{\rm SC}\sim 2.1\pm
0.5\times 10^{4}$ m$/$s, which is much smaller than $v_{\rm QH}$.
Theoretically, the edge state velocity renormalization for a non ideal
interface can be derived from the Andreev bound states spectrum[39] (see
Methods). From Eq. (2) in Methods, we obtain $v_{\rm SC}\sim 6\times 10^{4}$
m$/$s, whose order of magnitude is in good agreement with the experimental
estimate. Interestingly, such a renormalization leads to a suppression of
$v_{\rm SC}$ when $\Delta$ vanishes at the upper critical field of the
electrodes. This effect may be delicate to observe due to a crossover at
$E_{\rm Th}\gtrsim\Delta$ to a regime where normal electrons take over the
interference pattern.

The key parameter that limits the supercurrent in QH Josephson junctions at
finite temperature is therefore the width of the superconducting interface,
which severely reduces the Thouless energy via edge state velocity
renormalization[39, 21]. This could provide an explanation for measurable
supercurrent limited to small magnetic fields and high filling factors in the
large junctions of Ref.[8]. In contrast, in the graphene nanoribbons studied
here, a well-defined critical current withstands up to 8 T on the $h/2e^{2}$
QH plateau. Furthermore, contrary to conventional Josephson junctions,
increasing interface transparency further suppresses supercurrent by reducing
$v_{\rm SC}$ (see Eq. (2) in Methods) and consequently the Thouless energy.
Counter-intuitively, low transparency contacts may thus be a way to enhance
critical current.

For consistency check, we studied wider junctions with $W=2.3$ to
$2.4\,\upmu$m and $L=100$ to $300$ nm on sample HV88 with similar contact
transparencies (see HV88-G and HV88-H in SI Table S1). In these devices, the
supercurrent is absent on QH plateaus and mostly visible at the transition
between plateaus, where transport takes place through the bulk of the junction
(see Extended Data Fig. 8). Note that other mechanisms involving decoherence
due to disorder at the interface[22, 23] or vortices[46, 24] could be equally
relevant and lead to the same conclusion, although more delicate to assess
experimentally[16].

Inspecting the current-voltage characteristics at higher bias, we further
observe signatures of CAES interference along the superconducting interface,
which emerge as filling factor dependent resistance fluctuations[47, 48].
Figure 4d and e display the differential resistance fluctuations related to
the background fluctuations of the switching current and resistance seen in
Figs. 2b and d. Those are strongly suppressed above a voltage bias of about
$1.4$ mV, which is close to $2\Delta/e$ as usually expected in standard hybrid
Josephson junctions[49]. The fluctuation pattern echoes recent conductance
fluctuations observed downstream superconducting interfaces[12], ascribed to
CAES interference involving intervalley-scattering[50].

Interestingly, Fig. 4d reveals the breakdown of the $h/2e^{2}$ QH plateau with
a drop in resistance at high current bias near the transition to the next QH
plateau (see black arrows). Although at $\nu=2$ this breakdown occurs here at
relatively large bias with respect to superconducting features, the situation
may be detrimental for other QH plateaus with small energy gaps as those of
the fractional QH states. For these correlated states, the need to reduce the
superconducting interface width to obtain a robust supercurrent should
therefore meet another constraint related to edge transport, since QH
breakdown critical current is significantly weakened in narrow structures.

Finally, we shall recall the spurious charging effects in Fabry-Pérot QH
interferometers that have competed with and obscured the Aharonov-Bohm effect
for some time[51]. It is therefore appropriate to examine the role of the
Coulomb interaction in our small devices. As shown in Extended Data Figure 9,
we observed Coulomb blockade signatures at the edges of the QH plateau and in
the transition to the next plateau, in the regime where a compressible island
should form in the bulk. However, the Coulomb blockade quickly disappears as
soon as the QH plateau is entered to give way to Aharonov-Bohm interference.
Furthermore, the slope of the interference fringes in the $\nu-B$ plane is
negative, which is in agreement with Aharonov-Bohm dominated oscillations[51].
The distinct separation between Coulomb blockade signatures near the plateau
transition and Aharonov-Bohm interference on the plateau, along with the
negative fringes slope and correct area hence rule out charging effects in the
$2\phi_{0}$-periodic oscillations.

### Methods

#### Sample fabrication and parameters

The graphene and hBN flakes used to fabricate the heterostructures were
exfoliated and selected according to their surface cleanliness as determined
by atomic force microscopy (AFM). AFM images were further used to locate the
nanoribbons for subsequent lithography steps, and to measure the nanoribbon
width at different positions with an accuracy of about 30 nm, as shown in
Extended Data Fig. 1. Graphene/hBN heterostructures were assembled with the
van der Waals pick-up technique[40] using a polycarbonate polymer. The
heterostructures were deposited on a thin layer of graphite serving as a back-
gate electrode. Large scale wiring (Ti/Au bilayer) and superconducting MoGe
electrodes were patterned by two consecutive fabrication sequences including
electron-beam lithography, metal deposition and liftoff. MoGe electrodes were
deposited by DC-magnetron sputtering of a $\rm{Mo}_{50}\rm{Ge}_{50}$ target,
after etching of the stack with a CHF3/O2 plasma directly through the resist
pattern used to define the contacts. The critical temperature of MoGe tested
on a separate film is 5.9 K corresponding to a superconducting gap of
$\Delta=895\,\upmu$eV. The upper critical field of the MoGe electrodes of the
sample varies between 12.5 to 13 T (see Fig. S1). The geometrical parameters
of the two samples studied in this work, HV88 and DP24, are reported in
Extended Data Table 1. The top hBN, bottom hBN and graphite thicknesses are
12.7, 23.2 and 7.7 nm for HV88, and 25, 36 and 20 nm for DP24.

Each junction area $A$ is estimated from the superposition of the lithographic
design with the AFM image of the heterostructure as illustrated in Extended
Data Fig. 5, and confirmed with the AFM image of the device after contact
fabrication (see Extended Data Fig. 5f-h). The precision on the flake width
from the AFM image $\delta W_{\rm{AFM}}$ is estimated to be $\pm 30\ \rm{nm}$.
The junction width uncertainty $\delta W$ is dominated by this $\delta
W_{\rm{AFM}}$ with negligible contribution from lithographic alignment
uncertainty ($\delta W_{\rm{al}}$). Indeed, a lithographic misalignment
$\delta y$ along the junction width direction, estimated to be smaller than
100 nm, has no effect on the junction width since the electrodes are parallel
over more than 300 nanometers as shown in Extended Data Fig. 5a. The alignment
precision of the superconducting electrodes with respect to the flake in the
$x$ direction is estimated to be about $\delta x=\pm 130$ nm. The resulting
width uncertainty is given by $\delta W_{\rm{al}}=\delta x\sin{\theta}$ with
$\theta$ the nanoribbon opening angle as defined in Extended Data Fig. 5a.
With $\theta_{\rm{HV88}}=7.5^{\circ}$ and $\theta_{\rm{DP24}}=0.6^{\circ}$,
the width uncertainty from alignment $\delta W_{\rm{al}}$ is about $\pm 17\
\rm{nm}$ for HV88 and $\pm 1.4\ \rm{nm}$ for DP24. The total uncertainty on
the junction width $\delta W$ is thus given by $\delta W=\sqrt{\delta
W_{\rm{AFM}}^{2}+\delta W_{\rm{al}}^{2}}$, hence $\pm 35\ \rm{nm}$ for HV88
and $\pm 30\ \rm{nm}$ for DP24. The uncertainty on the junction length $\delta
L$ originating from lithographic development and etching is estimated from
dose test measurements to be about $\pm 20\ \rm{nm}$. The resulting
uncertainty in the junction area $\delta A=L\Delta W+W\delta L$ is reported
for all junctions in Extended Data Table 1.

####  .1 Measurements

Measurements were performed in a dilution refrigerator with a base temperature
of $0.01$ K and equipped with a 14 T superconducting solenoid magnet. The sub
nA switching current measurements (see Fig. 1) were resolved thanks to a
heavily filtered wiring comprising feedthrough $\pi$-filters (Tusonix
#4209-053) at room temperature, custom-designed all stainless steel pico-coax
cables (0.05 mm inner conductor diameter, 0.5 mm outer diameter), copper
powder filters at the mixing chamber plate, and 47 nF (NPO dielectric[52])
capacitor to ground on the sample holder. $I-V$ characteristics were measured
in a pseudo-four probe configuration as illustrated in Fig. 1a with an
acquisition card (National Instruments NI-6346). The voltage drop across the
junction was measured with a differential FET amplifier (DLPVA-100-F-D from
FEMTO Messtechnik GmbH). Oversampling at 30 to 150 kHz per point with the
acquisition card, along with very low level of low-frequency noise obtained by
ground loop mitigation, enabled us to obtain well defined $I-V$ curves with an
acquisition time of $1$ to $5$ s per curve. This fast $I-V$ acquisition
enabled us to measure more than 30 000 $I-V$ curves in the $V_{\rm g}-B$ plane
to entirely map out the QH plateau and to observe the critical current and
resistance oscillations. Differential resistance data were obtained by
numerically differentiating $I-V$ characteristics. Only the resistance data of
Fig. S13c and d were obtained by standard lockin amplifier technique. Shapiro
maps were obtained with a microwave radiation source feeding a coaxial cable
with the open end positioned about one millimeter above the devices.

####  .2 Edge state velocity renormalization

The energy spectrum of the Andreev bound states along a superconducting
interface has been calculated by Hoppe et. al.[39]. For a non-ideal interface,
it reads:

$E=\Delta\frac{(2N+1)\pi\pm\arccos{\Gamma_{0}}-2Xk_{\rm{F}}}{q+\pi\Delta/(\hbar\omega_{\rm{c}})},$
(1)

with $X$ the distance from the interface, $N$ the Landau level index, and
$\Gamma_{0}$ is an oscillatory function of the filling factor.
$q=2s/(1+s^{2}+w^{2})$ with $s=v_{\rm{N}}/v_{\rm{S}}$ the Fermi velocity
mismatch between graphene and the superconductor and
$w=k_{\rm{F}}U_{0}/E_{\rm{F}}$ is the scattering at the interface
corresponding to 2Z in the Blonder-Tinkham-Klapwijk model[53, 49]. While $s$
can be evaluated to be 1.4 (Fermi velocity is $v_{\rm{S}}=5.2\times
10^{5}\,\rm{m/s}$ in MoGe[54], and $v_{\rm{N}}=1\times 10^{6}\,\rm{m/s}$ in
graphene), there is no experimental way to assess $U_{0}$. However, using the
expression
$\mathcal{T}=\frac{v_{\rm{N}}v_{\rm{S}}}{(v_{\rm{N}}+v_{\rm{S}})^{2}/4+U_{0}^{2}}$
of the transparency for a non-perfect superconductor-metal interface of
Ref.[55], we can identify $q=\mathcal{T}/(2-\mathcal{T})$ with
$\mathcal{T}=4s/((1+s)^{2}+w^{2})$. Then, the edge velocity is given by
$v_{\rm{SC}}=-\frac{l_{B}^{2}}{\hbar}\frac{\partial E}{\partial X}$, which,
with Eq. (1), leads to:

$v_{\rm{SC}}=\frac{2v_{\rm{N}}}{\pi+q\frac{\hbar\omega_{\rm{c}}}{\Delta}}.$
(2)

The contact transparency $\mathcal{T}=0.5$ (see SI table S1) gives $q=0.3$.
With a superconducting critical temperature $T_{\rm{c}}=5.9$ K for MoGe (see
SI) corresponding a superconducting gap
$\Delta_{0}=1.76k_{\rm{B}}T_{\rm{c}}=895\ \upmu\rm{eV}$, we estimate
$\Delta=\Delta_{0}\sqrt{1-(H/H_{\rm{c2}})^{2}}=820\,\upmu\rm{eV}$ at 5 T, with
$H_{\rm{c2}}=12.5$ T (see SI). With $\hbar\omega_{\rm{c}}=81\rm{meV}$ at 5 T,
we obtain $v_{\rm{SC}}=6\times 10^{4}\ \rm{m/s}$.

####  .3 Coulomb blockade at the plateau transition

The Coulomb blockade, an ubiquitous phenomenon in small mesoscopic systems, is
found to occur at the transition between the plateaus. To illustrate this
effect, we focus in Extended Data Fig. 9 on the left side of the $h/2e^{2}$
plateau upon approaching the transition to $h/e^{2}$ plateau, at $B=14$ T,
that is, above the critical field of the superconducting electrodes. A clear
transition from the Fabry-Pérot checkerboard pattern, for $V_{\rm g}>0.6$ V,
to Coulomb diamonds, for $V_{\rm g}<0.6$ V, occurs (see Extended Data Fig.
9b). Note that we have continuously observed $h/e$-periodic resistance
oscillations beyond the critical field of the electrode, but with much lower
visibility and higher Thouless energy. Contrary to the orthodox picture, the
Coulomb blockade leads to quantized conductance inside the diamond and a
decrease in the conductance at resonance (see diamonds in Extended Data Fig.
9b delimited by darker conductance lines). Physically, as the back-gate
voltage is decreased near the plateau transition, charge carriers are
progressively removed from the zeroth Landau level, resulting in the emergence
of a compressible island in the graphene bulk (see Extended Data Fig. 9c). As
a result, this island, when tunnel-coupled to the QH edge channels on both
sides of the sample, induces back-scattering at resonance, hence decreasing
the QH conductance.

From the Coulomb diamonds in Extended Data Fig. 9b, we extract a charging
energy of $1.8$ meV as well as an estimate of the diameter of the compressible
island of $\sim 140$ nm, assuming it to be circular, which is a reasonable
size for the dimension of the junction ($L\times W=200\times 125$ nm2). This,
together with the remarkable regularity of the diamonds, confirm the picture
of a single and large compressible island in the graphene bulk upon
approaching the transition between plateaus.

### Acknowledgments

We thank D. Basko, C. Beenakker, M. Feigel’Man, L. Glazman, M. Houzet, V.
Kurilovich, J. Meyer, Y. Nazarov, K. Snizhko and A. Stern for valuable
discussions. We thank F. Blondelle for technical support on the experimental
apparatus. Samples were prepared at the Nanofab facility of the Néel
Institute. This work has received funding from the European Union’s Horizon
2020 research and innovation program under the ERC grant SUPERGRAPH No.
866365. It also benefited from a French government grant managed by the ANR
agency under the ’France 2030 plan’, with reference ANR-22-PETQ-0003. B.S.,
H.S. and W.Y. acknowledge support from the QuantERA II Program that has
received funding from the European Union’s Horizon 2020 research and
innovation program under Grant Agreement No 101017733. K.W. and T.T.
acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233
and 23H02052) and World Premier International Research Center Initiative
(WPI), MEXT, Japan.

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Device | $L$ (nm) |  $W$ (nm) | $A\times 10^{3}\ (\upmu m^{2})$ | $C_{\rm{g}}\ (\rm{mF}/\rm{m}^{2})$
---|---|---|---|---
HV88-B | 140 | 178 | $25\pm 8$ | 1.27
HV88-C | 170 | 210 | $36\pm 9$ | 1.39
HV88-D | 200 | 247 | $49\pm 11$ | 1.43
HV88-E | 240 | 288 | $69\pm 13$ | 1.14
HV88-F | 270 | 334 | $90\pm 14$ | 1.55
HV88-G | 107 | 2332 | $250\pm 50$ | 2.12
HV88-H | 202 | 2434 | $492\pm 56$ | 1.93
DP24-C | 170 | 125 | $21\pm 8$ | 1.12
DP24-D | 200 | 125 | $25\pm 8$ | 1.89
Table 1: Device parameters. $L$ is the length separating the junction
electrodes and $W$ the width of the junction (see schematics in Fig. 5a). $A$
is the geometric area of the graphene (see Methods for uncertainty
calculation). $C_{g}$ is the capacitance per unit area deduced from the slope
of the $\nu=2$ quantum Hall plateau center (see SI section II.C).  Figure 5:
hBN-encapsulated graphene nanoribbon samples. a, Schematic describing the
uncertainty in the junction width $\delta W$ (red lines) and length $\delta L$
(vertical black dashed lines in the zoom). Horizontal black dashed lines
indicate the uncertainty in the graphene width $\delta W_{\rm{AFM}}$ from AFM
imaging. Blue rectangles represent MoGe electrodes correctly aligned, dashed
blue lines indicate the MoGe electrode position with a misalignment $(\delta
x,\delta y)$, which results in an uncertainty $\delta W_{\rm{al}}$ on the
graphene edges position. b, AFM picture for sample HV88, the encapsulated
graphene is readily visible with contrast enhancement. c, Zoom on the graphene
narrow part in (b), with the lithography pattern overlayed (Ti/Au lines in
orange and MoGe electrodes in blue). Red lines indicate the total width
uncertainty $\pm\delta W/2$ around the graphene edges. d, AFM picture of the
graphene nanoribbon of sample DP24 before hBN encapsulation. Inset: higher
resolution AFM picture. The arrows indicate the nanoribbon width of 125 nm. e,
AFM picture of sample DP24 after hBN encapsulation. The yellow arrows indicate
the nanoribbon position. The lithography pattern is overlayed on the picture.
The scale bars are 5 $\upmu$m (b), 400 nm (c), 2 $\upmu$m (d), 500 nm (d)
inset, 500 nm (e). f, AFM picture of sample DP24 contact fabrication. The
scale bar is 1 $\upmu$m. g, h, Height profiles along the blue and green lines
in the AFM image in f. The excess thickness on the contact edge results from
the liftoff process of the MoGe. It provides a good estimate of the position
of the contact to graphene.

Figure 6: Shapiro steps of the chiral supercurrent. $I-V$ characteristics
measured at 3.45 T on sample HV88-C, under microwave radiation of frequency
$f=1$ GHz. The $I-V$ curves are extracted from the data of Fig. 1e at a
microwave power of -13.8 dBm. The color-coded arrows indicate the current
sweep direction.

Figure 7: Cherkerboard patterns. a-c, Differential resistance modulation with
magnetic field and measured voltage showing a characteristic checkerboard
pattern. d-f, Fourier transform amplitude at the checkerboard oscillation
frequency as a function of the measured voltage. The orange curve is a fit to
the data following the same procedure as in Ref.[44]. a,d, Junction DP24-D
with fit parameters $E_{\rm{Th}}=640\,\upmu\rm{V}$ and $x=0$. b,e, Junction
HV88-C with fit parameters $E_{\rm{Th}}=240\,\upmu\rm{V}$ and $x=0.02$. c,f,
Junction HV88-F with fit parameters $E_{\rm{Th}}=140\,\upmu\rm{V}$ and
$x=0.02$.

Figure 8: Wide graphene Josephson junction. a-d, Top panels show differential
resistance maps as a function of back-gate voltage and dc current bias of
sample HV88-G ($W=2.3\,\upmu$m, $L=107$ nm). Bottom panels are linecuts of the
differential resistance at dc current biases of 0 nA and 26 nA, which show the
emergence of supercurrent pockets (zero resistance reached by the blue curve),
and the corresponding resistive state (yellow curve). The magnetic field is
indicated in each top panel. Supercurrent is visible only when the resistance
is not quantized, that is, at filling factors of QH plateaus that are not
developed, or in between plateaus. When a QH plateau emerges, as for instance
the $h/2e^{2}$ plateau for $B\geq 4$ T or the $h/6e^{2}$ plateau for $B\geq 6$
T, the supercurrent vanishes. This weakness is even more marked in sample
HV88-H shown in SI, which is twice longer. Note that the oscillatory behavior
of the resistance (see red curve at 26 nA) is characteristic of QH devices in
two-terminal configuration with $L\ll W$, see Ref.[41, 42].

Figure 9: Coulomb blockade on the QH plateau edge. a, Differential resistance
as a function of back-gate voltage for different dc current biases at $B=14$ T
on sample DP24-D. At low current, Coulomb diamonds appear as high frequency
and high amplitude oscillations visible at the transition to the $\nu=1$
plateau. $2\phi_{0}$-quantum interference gives rise to oscillations with
larger period and a very small amplitude on the plateau. The vertical dash
line indicates the limit of Coulomb diamonds seen in (b). The horizontal
dashed line indicates the resistance value of $h/2e^{2}$. b, Differential
conductance as a function of back-gate voltage and applied voltage across the
junction. Measurements are performed in voltage bias configuration. The
Coulomb diamonds disappearance coincides with the start of the checkerboard
pattern of the Aharonov-Bohm interference. c, Schematics of the scattering
process through a compressible island in the bulk[51], with the back-gate
voltage corresponding to the edge of the plateau, leading to the Coulomb
diamonds in (b).

##  Supplementary Information

###  I Critical temperature and upper critical field of the MoGe electrodes

We present in this section the superconducting properties of the MoGe
electrodes. We measured a superconducting transition temperature of $5.9$ K on
a separate bare MoGe film (see Fig. S1a.) deposited in the same conditions as
the electrodes of the samples. Transitions to normal state with magnetic field
for some MoGe electrodes of both samples HV88 and DP24 are shown in Fig. S1b.
The resistance was measured in a two-terminal configuration with a lock-in
amplifier technique, at a temperature of 0.05 K. A constant resistance of 1.06
$\rm{k}\Omega$ was subtracted to account for the wiring resistance in series
with the electrodes. We measured a critical field of about 12.5 T for HV88 and
13.5 T for DP24.

Figure S1: MoGe superconducting transition. a, Four-terminal resistance of a
MoGe thin film as a function of temperature. b, Two-terminal resistance of
superconducting electrodes on sample HV88 and DP24, a constant wiring
resistance was subtracted. The legend indicates the measured MoGe electrodes
(e.g. HV88-EF corresponds to the common electrode shared by junctions HV88-E
and HV88-F).

###  II Device characterization

Here, we present the characterization of all devices, including their normal
state properties at 6 K, the quantum Hall regime, and the Josephson junctions
properties at zero and low magnetic field.

####  II.1 Graphene nanoribbon devices

Figure S2 displays the resistance versus back-gate voltage measured at a
temperature of 6 K for HV88 and 4 K for DP24. The high value of resistance
results mainly from the contact resistance, which is high due to the very
narrow width of the contacts. The charge neutrality points of the devices are
located at low negative back-gate voltage, indicating a residual electron-type
doping. The higher value of the resistance for negative back-gate voltage is
characteristic of charge transfer from the contacts, which induces an electron
doping and thus a bipolar npn regime with reduced transparency. This bipolar
npn regime is consistent with the resistance fluctuations seen in all
junctions, which we ascribe to the Fabry-Pérot interferences between the pn
interfaces routinely observed in high mobility devices [56, 6, 57]. The
electron doping from the MoGe electrodes also explains why the quantum Hall
effect is not well defined on the hole side (negative back-gate voltage), as
shown in section II.2.

Figure S2: Normal state resistance of the devices. Pseudo four-terminal
resistance of the devices as a function of back-gate voltage, measured at
$B=0$ T.

####  II.2 Quantum Hall effect in the graphene nanoribbons

The quantum Hall effect was characterized with a dc current bias of $I=1.5\
\mu\rm{A}$ for HV88 at 10 mK and with a dc voltage bias of $V=2.5\ \rm{mV}$
for DP24 at 4 K, to prevent deviation from quantization due to sub-gap
features [58]. The resistance maps of the different junctions as a function of
back-gate voltage and magnetic field are shown in Fig. S3. As discussed in the
previous section, the hole side is badly quantized due to the npn bipolar
regime. On the electron side, the $\nu=2$ quantum Hall plateau is well
quantized from low magnetic field (from $\sim 1.5$ T depending on the device
for narrow junctions). The visibility of other QH plateaus with smaller energy
gap, as for instance the $\nu=1$ state, requires that the critical current for
their QH breakdown is larger than the non-linearities induced by the Andreev
reflections at finite bias (typically up to $2\Delta/e$). As a result, the
$\nu=1$ state emerges as a well quantized plateau from $B\sim 6$ T. It is
clearly visible in the line-cuts taken at $B=14$ T in Fig. S3. Plateaus for
filling factors $\nu>2$ meet the same issues. Their absence can furthermore be
accounted for by a lack of equilibration between the edge channel of the
zeroth Landau level and the edge channels of the higher Landau levels in our
mesoscopic devices [59].

The line-cuts of Fig. S3 measured with a small current excitation $I=10\
\rm{nA}$ at 14 T, that is, above the upper critical field of the electrodes
are not altered by superconducting sub-gap features nor quantum Hall
breakdown. We can see in Fig. S3e and d the emergence of the $\nu=3$ and 4
broken-symmetry states.

Figure S3: Quantum Hall effect characterization. Measurements on sample HV88
junction B to F in a-e and on sample DP24 junctions C and D in f-g. The left
panel is the junction differential resistance measured with a current of $1.5\
\mu\rm{A}$ at a temperature of 10 mK (except for DP24-C and D measured with a
bias voltage of 2.5 mV at 4 K) as a function of back-gate voltage and magnetic
field. The colormaps display the usual fan observed in the quantum Hall
regime. The right panel corresponds to separate measurements with smaller
current excitations at 14 T and 10 mK. Several broken-symmetry states emerge
when measured with a current of 10 nA, while a current of 75 nA is sufficient
to significantly reduce these plateaus widths.

####  II.3 Back-gate capacitance estimation

The quantum Hall data of Fig. S3 are used to quantify the back-gate
capacitance. While a two-plate capacitor model gives $C=1.4\
\rm{mF}/\rm{m}^{2}$ for sample HV88 and $C=0.89\ \rm{mF}/\rm{m}^{2}$ for
sample DP24 using a dielectric constant of $\varepsilon_{\rm{BN}}=3.3$ for
boron nitride [60], in very small structure the capacitance can be modified
due to screening by the electrodes. We thus estimated the back-gate
capacitance through the dispersion of the filling factor 2 plateau as shown in
Fig. S4. The middle of the plateau is defined as $\nu=2.0$. The curve at 160
nA is chosen as a reference to detect the plateau edges as it shows limited
deviation from quantization compared to both lower and higher currents. These
deviations originate from the non-linearities of the superconducting interface
at low current, and from the quantum Hall breakdown at high current. The
resulting capacitances that we used to calculate the filling factor are
reported in the Extended Data Table I.

Figure S4: Filling factor 2 plateau characterization in sample HV88-C. a-b,
Differential resistance at 14 T and 5.3 T at two different dc bias currents
(160 nA and 1.2 $\mu\rm{A}$), blue (purple) dots delimit the plateau edge
defined by a deviation of 2% from the plateau resistance. c, Detected plateau
edge and plateau center evolution with magnetic field.

####  II.4 Contact resistance

If one assume ballistic transport within the junction, we can estimate the
graphene resistance in the pure quantum limit [6]: $R_{Q}=h/e^{2}g_{s}g_{v}N$,
where $g_{s}=g_{v}=2$ account for the spin and valley degeneracies,
respectively, and N is the number of transverse modes propagating in the
graphene $N=k_{F}W/\pi$. $k_{F}=\sqrt{\pi n}$ and the electron density is
defined as $n=C(V_{g}-V_{CNP})/e$.

The difference between the measured resistance $R$ and $R_{Q}$ provides the
contact resistance $R_{c}=(R-R_{Q})/2$. At $V_{g}=1$ V, we obtain $460\pm 21$
$\Omega.\mu m$ averaged over 13 devices for the MoGe contacts. This
corresponds to an average transmission probability $Tr=R_{Q}/(R_{Q}+R_{c})\sim
0.23$. This transmission probability is half that obtained from the IV curves
in the superconducting state (see section II.E). Note that a similar, though
less pronounced, difference was observed in [6].

####  II.5 Supercurrent characterization at $B=0$ T

Figure S5 displays the differential resistance as a function of dc current
bias and back-gate voltage at zero magnetic field for all devices. The
decrease of resistance with back-gate voltage leads to an increase of
switching current as expected for a constant $I_{c}R_{\rm{n}}$ product, where
$R_{\rm{n}}$ is the normal state resistance. We obtained $I_{c}R_{\rm{n}}$
ranging from $\sim 87\,\mu$V to $\sim 243\,\mu$V.

Figure S5: Supercurrent at zero field and 10 mK. Differential resistance
versus back-gate voltage and dc current bias for sample HV88, junctions B to H
in a-g, and for junction DP24-C in h.

We estimated the contact transparency from $I-V$ characteristics following
Ref. [58, 61, 62]. The normal state resistance $R_{\rm{n}}$ is estimated from
a linear fit at high current bias (see Fig.S6), and the excess current
$I_{\rm{exc}}$ is defined as the intercept of this fit at zero voltage. The
conductance plotted as a function of measured voltage exhibits two maxima,
symmetric with respect to zero voltage, that we ascribe to twice the
superconducting gap $2\Delta$. Eq. (25) in Ref. [62] is used to deduce the
Blonder-Tinkham-Klapwijk parameter $Z$, which leads to the transparency
defined as $\mathcal{T}=1/(1+Z^{2})$. The transparency of our junctions are
reported in Table 2 for 7 junctions, along with $R_{\rm{n}}$, $I_{\rm{exc}}$,
and $\Delta$.

Figure S6: Excess current and superconducting gap of the MoGe electrode. a, $I-V$ characteristic of the junction HV88-F at $B=0\ \rm{T}$ and $V_{\rm{g}}=1\ \rm{V}$. The dashed line is a linear fit of the high current bias part. The intercept of this fit with the x-axis gives the excess current $I_{\rm{excess}}$ (see grey arrow). Inset: Higher resolution $I-V$ measurement at low current bias showing more clearly the supercurrent. b, Numerical derivative $dI/dV$ of a versus measured voltage. The voltages of the two conductance maxima are associated with $\pm 2\Delta/e$ according to Ref. [58]. Junction |  $V_{\rm{CNP}}$ (V) |  $V_{\rm{CNP}}^{\nu=2}$ (V) | $V_{\rm{g}}\ (\rm{V})$ | $\Delta\ (\mu\rm{eV})$ |  $I_{\rm{exc}}$ (nA) |  $R_{\rm{n}}$ ($k\Omega$) | $T$ |  $R_{\rm{n}}I_{\rm{c}}$ ($\mu\rm{V}$)
---|---|---|---|---|---|---|---|---
HV88-C | -0.13 | -0.14 | 1 | 852 | 49 | 4.5 | 0.49 | 150
HV88-D | -0.19 | -0.15 | 1 | 865 | 46 | 3.8 | 0.47 | 91
HV88-E | -0.14 | -0.25 | 1 | 876 | 86 | 3.0 | 0.5 | 96
HV88-F | -0.04 | -0.08 | 1 | 897 | 59 | 3.2 | 0.47 | 87
HV88-G | -0.52 | -0.24 | 2 | 762 | 552 | 0.34 | 0.48 | 146
HV88-H | -0.34 | -0.23 | 2 | 773 | 434 | 0.34 | 0.47 | 137
DP24-C | -0.4 | -0.12 | 1 | 754 | 83 | 4.2 | 0.56 | 243
Table 2: Junction characteristics at $B=0\,\rm{T}$. Charge neutrality point
positions $V_{\rm{CNP}}$ were extracted from the Dirac peak positions in plots
of the resistance as a function of the gate-voltage (see Figure S2). We also
provide the charge neutrality point positions $V_{\rm{CNP}}^{\nu=2}$ estimated
from the fits to the $\nu=2$ plateau center used for capacitance estimation
(see section II.3). Note that filling factor evaluation was done using
$V_{\rm{CNP}}^{\nu=2}$ for the Dirac point gate voltage value. The back-gate
voltage value at which the gap $\Delta$ of the superconducting electrodes, the
junctions excess current $I_{\rm{exc}}$, the normal state resistance
$R_{\rm{n}}$, the graphene-superconducting electrode interface transparency
$T$ as well as the $R_{\rm{n}}I_{\rm{c}}$ products were evaluated is specified
in the $V_{\rm{g}}$ column.

####  II.6 Fraunhofer patterns

In this section we present the quantum interference (Fraunhofer) patterns of
the Josephson junctions at low magnetic field. Figure S7 displays the
differential resistance as a function of current bias and magnetic field for 6
devices. The evolutions of the switching current do not follow the usual
Fraunhofer pattern with a sinus cardinal decay. This is not clearly understood
but we cannot exclude vortices trapped in the superconducting electrodes due
to the relatively large field needed for reaching a flux quantum through the
small graphene area of the junctions (e.g. $B=83$ mT for device HV88-B), as
well as deviation from the Fraunhofer pattern due to the square geometry of
the graphene [63]. Still, we can extract a flux period from the central lobs
of these quantum interference patterns. The magnetic field at which the
switching current vanishes for the central lob corresponds theoretically to
$\phi_{0}=h/2e$ [64]. We thus added white dashed lines in Fig. S7 that
indicates the equivalent magnetic field for one flux quantum through the
graphene area. We obtain a remarkably good agreement for all devices.
Interestingly, Fraunhofer patterns nearly matching the graphene area have been
observed in similar devices with square geometry ($W\sim L$) in [56].

Figure S7: Quantum interference of the critical current. Measurements on
sample HV88 junction B to E in a-d and on sample DP24 junctions C and D in
e-f. a-f, Differential resistance versus magnetic field and dc current bias.
The supercurrent delimiting zero (in black) and finite resistance state (in
yellow and purple) shows oscillatory behavior resembling Fraunhofer pattern.
The white dashed lines indicate the expected periodicity for a superconducting
flux quantum $\phi_{0}=h/2e$ through the graphene area.

###  III Additional Shapiro maps

We show in this section additional Shapiro maps, complementing the data of
Fig. 1e. Figure S8 displays the Shapiro maps of the differential resistance
along with $I-V$ characteristics measured with the same parameters of magnetic
field and back-gate voltage as in Fig. 1e but with microwave frequencies of 3
GHz, in Fig. S8a and c, and at 6 GHz in Fig. S8b and d.

Figure S8: Additional Shapiro maps. a-b, Differential resistance as a function
of current bias (sweep up) and microwave power at 3 GHz (a) and 6 GHz (b),
measured in junction HV88-C with the same parameters of magnetic field and
back-gate voltage as in Fig. 1e. c-d, $I-V$ curves at a fixed microwave power
indicated by the white lines in a and b, showing successive Shapiro voltage
steps of height $hf/2e$. The blue $I-V$ is sweep up and purple is sweep down.

###  IV $2\phi_{0}$-periodic oscillations

####  IV.1 Constant back-gate voltage versus constant filling factor

In this section we illustrate the relevance of studying the flux periodicity
of the supercurrent and resistance oscillations at constant filling factor
compared to constant back-gate voltage. Figure S9 displays the zero bias
differential resistance of sample HV88-D as a function of back-gate voltage
and magnetic field. The measurement is restricted to the $\nu=2$ QH plateau to
limit the the data acquisition to a reasonable duration. Fringes similar to
that of Fig. 2d are visible in this resistance map. However, extracting a
line-cut at constant back-gate voltage (along the vertical dashed line at
$V_{\rm g}=0.1$ V) does not exhibit any oscillatory behavior, as shown in Fig.
S9b. On the other hand, a line-cut at constant filling factor (along the
diagonal dashed line at $\nu=1.8$) shown in Fig. S9c clearly unveils the
$2\phi_{0}$-periodic oscillations.

Figure S9: Absence of oscillation with back-gate voltage. a, Differential
resistance map as a function of back-gate voltage and magnetic field at zero
current bias in sample HV88-D. b, Resistance versus magnetic field at a fixed
back-gate voltage of $V_{\rm{g}}=0.1\ \rm{V}$ (taken along the vertical dashed
line in (a)) after background subtraction. c, Resistance versus magnetic field
at a fixed filling factor $\nu=1.8$ (taken along the diagonal dashed line in
(a)) after background subtraction.

####  IV.2 Switching current and resistance oscillations on additional devices

We show in Fig. S10 additional measurements of the $2\phi_{0}$-periodic
oscillations of the supercurrent for the junctions HV88-B, HV88-C and HV88-D.
The differential resistance map is plotted as a function of dc current bias
and magnetic field, at constant filling factor. The red traces superimposed on
each Figures show the detected switching current. Note that such a detection
was used for the switching current map in Fig 2c for HV88-B.

Figure S10: Critical current oscillation with magnetic field in sample HV88.
a, Junction B. b, Junction C. c, Junction D. Differential resistance as a
function of current bias and magnetic field at constant filling factor. The
extracted switching current is superimposed as a red line.

In Fig. S11 we show the switching current map and the zero bias resistance as
a function of filling factor and magnetic field for the junctions HV88-C and
HV88-D. Fig. S11c and g display the Fourier transform of the resistance map as
a function of frequency $1/\Delta B$ and filling factor for HV88-C and HV88-D,
respectively. Fig. S11d and h display the Fourier transform of the critical
current oscillations as a function of frequency $1/\Delta B$ for HV88-C and
HV88-D, respectively. The frequencies of these oscillations are in excellent
agreement with the graphene area corrected by the magnetic length, therefore
confirming the results presented in the main text.

The data of the two longest and widest nanoribbons, HV88-E, HV88-F, are shown
in Fig. S12. Despite a smaller magnetic field period, we resolve the
supercurrent and resistance oscillations on the QH plateau. Differential
resistance maps as a function of magnetic field and current bias at fixed
filling factors in Fig. S12a and b clearly show the $2\phi_{0}$-periodic
oscillations of the supercurrent and of the resistance, as indicated by the
white dashed lines. Similar results have been obtained on junctions DP24-C and
DP24-D, as shown in Fig. S13.

Figure S11: $2\phi_{0}$ oscillations of critical current and resistance in
junctions C and D of sample HV88. a-d, Junction HV88-C. e-h, Junction HV88-D.
a, e, Critical current and b, f, differential resistance at zero bias are
periodically modulated with both magnetic field and filling factor. c, g, The
normalized Fourier transform of the resistance at finite current (5 nA) shows
a small increase of the oscillation frequency with filling factor. In c and g,
the vertical lines correspond to the graphene area (right line) while the
dashed lines correspond to the graphene area corrected as
$(L-2l_{B})$x$(L-2l_{B})$ (left line). The magnetic length $l_{B}$ is taken at
$4.75$ T (resp. $5$ T) in (c) (resp. g). The Fourier transform shown in (c)
was obtained in a magnetic field window ranging from $2.5$ T to $7$ T. The
Fourier transform shown in (g) was obtained in a magnetic field window ranging
from $3$ T to $7$ T. The error bar corresponds to the uncertainty on the
graphene area detailed in Methods. d, h, The Fourier transform of the critical
current shows a peak at a frequency compatible with the expected one. The
colored rectangle is centered at the frequency corresponding to $2\phi_{0}$
oscillations for the graphene area, its width is given by the uncertainty on
this area. The Fourier transform signal in (d) (resp. (h)) was obtained in a
magnetic field window going from $2$ T to $7$ T (resp. from $1$ to $8$ T).
Figure S12: $2\phi_{0}$ oscillations of critical current and resistance in
junctions E and F of sample HV88. a, c, Junction HV88-E. b, d, Junction
HV88-F. a-b, Critical current and differential resistance are periodically
modulated with magnetic field. In a, the left (resp. right) white comb
indicates the $2\phi_{0}$ magnetic field periodicity corresponding to $113\%$
(resp. $89\%$) of the graphene area. In b, the left (resp. right) white comb
indicates the $2\phi_{0}$ magnetic field periodicity corresponding to $99\%$
(resp. $90\%$) of the graphene area. c-d, The differential resistance at 0 nA
shows oscillations with filling factor and magnetic field. Figure S13:
$2\phi_{0}$ oscillations of critical current and resistance in junctions C and
D of sample DP24. a-b, Differential resistance vs current bias and magnetic
field at a constant filling factor for junction DP24-C in a and for junction
DP24-D in b. $2\phi_{0}$ oscillations of resistance and critical current are
visible, the dashed lines indicating the theoretical period within the device
area uncertainty, that is $125\%$ of the device DP24-C graphene area in (a)
and $123\%$ of the device DP24-D graphene area in (b). c-d, Differential
resistance mapping as a function of filling factor and magnetic field for
junction DP24-C in (c) and for junction DP24-D in (d) with a current bias of 1
nA. e, Fourier transform of the resistance oscillations at a fixed filling
factor showing that the oscillation period correspond to a flux quantum of
$2\phi_{0}$. Bottom curve DP24-C (obtained at $\nu=3$), top curve DP24-D
(obtained at $\nu=1.44$).

###  V Additional wide junction

We present in this section the data for a second wide junction (HV88-H) in
complement to Extended Data Fig. 4. This junction has nearly the same width
($W=2.4\,\mu$m) but is twice longer than junction HV88-G. Figure S14 displays
the differential resistance as a function of back-gate voltage and dc current
bias at different magnetic field values, along with the differential
resistance measured at 0 and 26 nA to show the superconducting pocket and the
resistive state. As in Extended Data Fig. 4, supercurrent is conspicuously
absent on the $h/2e^{2}$ QH plateau. For low magnetic field there is a finite
supercurrent in the $h/6e^{2}$ QH plateau. However, this supercurrent vanishes
when the resistance becomes quantized upon increasing magnetic field,
indicating that this junction cannot maintain supercurrent in the edge
transport regime. This is consistent with the suppression of the Thouless
energy by the long superconducting interfaces, which in turn suppress the
supercurrent [21].

Figure S14: Additional wide graphene Josephson junction. a-d, Top panels show
differential resistance maps as a function of back-gate voltage and dc current
bias for device HV88-H ($W=2.4\,\mu$m, $L=202$ nm) at $B=2$ T, $3$ T, $4$ T
and $5$ T. Bottom panels are linecuts of the differential resistance at dc
current biases of 0 nA and 16 nA, which show the emergence of supercurrent
pockets (zero resistance reached by the blue curve), and the corresponding
resistive state (orange curve). The magnetic field is indicated in each top
panel. Supercurrent is visible only when the resistance is not quantized, that
is, at filling factor of QH plateau that are not developed, or in between
plateaus. When QH plateau emerges, as for instance the $h/2e^{2}$ plateau for
$B\geq 3$ T or the $h/6e^{2}$ plateau for $B\geq 3$ T, the supercurrent
vanishes. Note that the oscillatory behavior of the resistance (orange curve)
is characteristic of QH devices in two-terminal configuration with $L\ll W$
[41, 42].