Present address: ]Institute of Physics, Czech Academy of Sciences, 162 00
Prague, Czech Republic

# Flux-Tunable Josephson Diode Effect in a Hybrid Four-Terminal Josephson
Junction

M. Coraiola  IBM Research Europe—Zurich, 8803 Rüschlikon, Switzerland  A. E.
Svetogorov  Fachbereich Physik, Universität Konstanz, D-78457 Konstanz,
Germany  D. Z. Haxell  IBM Research Europe—Zurich, 8803 Rüschlikon,
Switzerland  D. Sabonis  IBM Research Europe—Zurich, 8803 Rüschlikon,
Switzerland  M. Hinderling  IBM Research Europe—Zurich, 8803 Rüschlikon,
Switzerland  S. C. ten Kate  IBM Research Europe—Zurich, 8803 Rüschlikon,
Switzerland  E. Cheah  Laboratory for Solid State Physics, ETH Zürich, 8093
Zürich, Switzerland  F. Krizek  [  IBM Research Europe—Zurich, 8803
Rüschlikon, Switzerland  Laboratory for Solid State Physics, ETH Zürich, 8093
Zürich, Switzerland  R. Schott  Laboratory for Solid State Physics, ETH
Zürich, 8093 Zürich, Switzerland  W. Wegscheider  Laboratory for Solid State
Physics, ETH Zürich, 8093 Zürich, Switzerland  J. C. Cuevas  Fachbereich
Physik, Universität Konstanz, D-78457 Konstanz, Germany  Departamento de
Física Teórica de la Materia Condensada and Condensed Matter Physics Center
(IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain  W. Belzig
Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany  F.
Nichele<EMAIL_ADDRESS>IBM Research Europe—Zurich, 8803 Rüschlikon,
Switzerland

###### Abstract

We investigate the direction-dependent switching current in a flux-tunable
four-terminal Josephson junction defined in an InAs/Al two-dimensional
heterostructure. The device exhibits the Josephson diode effect, with
switching currents that depend on the sign of the bias current. The
superconducting diode efficiency, reaching a maximum of $|\eta|\approx 34\%$,
is widely tunable—both in amplitude and sign—as a function of magnetic fluxes
and gate voltages. Our observations are supported by a circuit model of three
parallel Josephson junctions with non-sinusoidal current–phase relation. With
respect to conventional Josephson interferometers, phase-tunable multiterminal
Josephson junctions enable large diode efficiencies in structurally symmetric
devices, where local magnetic fluxes break both time-reversal and spatial-
inversion symmetries. Our work establishes a pathway to develop Josephson
diodes with wide-range tunability and that do not rely on exotic materials or
externally applied magnetic fields.

Nonreciprocal transport phenomena play a key role in modern electronics, with
semiconductor diodes serving as the fundamental components for numerous
devices [1]. In analogy to the semiconductor diode, whose electrical
resistance strongly depends on the current direction, a superconducting diode
allows a larger supercurrent to flow in one direction compared to the other
[2]. Nonreciprocal supercurrents were recognized already in the 1970s in
superconducting quantum interference devices (SQUIDs) based on superconducting
bridges [3] and tunnel Josephson junctions (JJs) [4, 5], arising as a
consequence of the finite loop inductance. Direction-dependent switching
currents were also observed in conventional superconducting thin films and
interpreted as a manifestation of microscopic asymmetries in the device
geometry [6]. More recently, the superconducting diode effect (SDE) has
sparked renewed interest, driven by its connection to fundamental properties
of a diverse range of superconducting systems, where the breaking of both
inversion and time-reversal symmetries is required for the effect to occur.
Since its observation in artificial superlattices [7], the SDE has been the
subject of thorough experimental and theoretical investigation, both in
junction-free thin films [8, 9, 10, 11, 12, 13] and JJs based on
semiconductors with spin–orbit coupling [14, 15, 16, 17], finite-momentum
superconductors [18, 19, 20] or multilayered materials, realizing sizeable
asymmetries even without external magnetic fields [21, 22, 23, 24, 25, 26, 27,
28].

An alternative platform proposed to achieve the SDE in Josephson devices—where
it is usually referred to as Josephson diode effect (JDE)—relies on a
supercurrent interferometer, where two JJs with nonsinusoidal current–phase
relations (CPRs) are combined in a SQUID [29, 30]. Such CPRs, containing
contributions from higher harmonics than the conventional $2\pi$-periodic
component, are routinely attained in high-quality superconductor–semiconductor
planar materials [31, 32, 33, 34], where hybrid JJs host Andreev bound states
(ABSs) characterized by high transmission. Key ingredients for the JDE to
occur in this system are the different harmonic content between the two JJs,
and a magnetic flux threading the SQUID loop [29], as recently demonstrated in
two-dimensional (2D) electron [35, 36] and hole [37] systems, obtaining large
diode efficiencies at equilibrium up to approximately $30\%$. Furthermore,
three-terminal Josephson devices without phase control were shown to realize
the JDE by exploiting high harmonic terms in the CPR [38], multiple bias
currents [39, 40] and Andreev molecules [41, 42].

In this work, we report on the superconducting transport properties of a
superconductor–semiconductor four-terminal JJ (4TJJ) embedded in a double-loop
geometry. Supercurrents are tuned by magnetic fluxes threading the
superconducting loops and by gate electrodes that control the number and
transmissions of ABSs. We find strong JDE, tunable in both efficiency and sign
with gate voltages and two independent superconducting phases—controlled via
integrated flux-bias lines, without the need for external magnetic
fields—reaching a peak efficiency of $\pm 34\%$. One of the loops was further
controlled by gating an additional JJ with large critical current, enabling
operation of the device in a single-loop configuration. We provide an in-depth
explanation of the JDE in our system by means of a simple circuit model, which
maps our device to the combination of two SQUIDs, or a bi-SQUID. Simulations
are performed both in an idealized case with minimal assumptions, and in an
extended version that accurately captures the experimental results. We
demonstrate several novelties with respect to previous work. First, we devise
a geometry that combines both gate control of the current path and phase
tunability in a 2D space, offering new insights into multiterminal JJs.
Second, the phase-tunable bi-SQUID allows for a large diode efficiency to
occur without the need of a JJ imbalance in harmonic content, which is
required in conventional interferometers [29]. Third, phase control enables
large tuning of the JDE in amplitude and sign, including a vanishing diode
efficiency in extended regions of the phase space. In the light of our
results, multiterminal JJs in superconductor–semiconductor hybrid systems
offer advantages that are pivotal for the realization of nonreciprocal
transport phenomena. Thanks to the nonsinusoidal CPR and the possibility to
break time-reversal symmetry solely by flux biasing, large and tunable diode
efficiencies are naturally obtained without the need of an external magnetic
field. Future work might further expand the study of multiterminal devices to
realize nonreciprocal transport in the linear regime [43, 44], presenting
opportunities for innovative applications.

Figure 1: Device under study and measurement setup. (a) False-colored scanning
electron micrograph of a device identical to that under study. Exposed III–V
semiconductor is represented in pink, Al in blue, gate electrodes in yellow
and flux-bias lines in purple. Bias current $I_{\mathrm{SD}}$, flux-line
currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, magnetic fluxes threading the
superconducting loops $\mathit{\Phi}_{\mathrm{L}}$ and
$\mathit{\Phi}_{\mathrm{R}}$, and gate voltages $V_{\mathrm{S}}$,
$V_{\mathrm{L}}$, $V_{\mathrm{M}}$, $V_{\mathrm{R}}$ and $V_{\mathrm{J}}$ are
labeled. Superconducting terminals S, L, M, R and the common node D are also
indicated. (b) Zoom-in of (a) in the vicinity of the four-terminal Josephson
junction. (c) Schematic representation of the device with the measurement
setup, using the same color labeling as in (a) and (b). Gate electrodes are
not shown. Figure 2: Phase-tunable Josephson diode effect. (a) Differential
resistance $R$ as a function of left flux-line current $I_{\mathrm{L}}$ and
source–drain bias current $I_{\mathrm{SD}}$, for fixed right flux-line current
$I_{\mathrm{R}}=0.1~{}\mathrm{mA}$. The map is obtained by merging two
datasets recorded with $I_{\mathrm{SD}}$ ramping from $0$ to either positive
or negative values (see white arrows). Switching current nonreciprocities are
highlighted by cyan annotations. A point where the switching current reaches
zero is indicated by the yellow arrow. Gate voltages were set to
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$,
$V_{\mathrm{S}}=0.1~{}\mathrm{V}$, $V_{\mathrm{M}}=-0.15~{}\mathrm{V}$ and
$V_{\mathrm{J}}=0$. (b, c) Switching currents $I^{+}_{\mathrm{sw}}$ and
$I^{-}_{\mathrm{sw}}$, measured for positive and negative $I_{\mathrm{SD}}$
respectively, as functions of $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. Arrows in
(b) indicate the directions along which magnetic fluxes threading the left and
right superconducting loop, $\mathit{\Phi}_{\mathrm{L}}$ and
$\mathit{\Phi}_{\mathrm{R}}$, vary independently. Each arrow represents the
addition of one superconducting flux quantum $\mathit{\Phi}_{0}$ to the
corresponding flux. (d) Superconducting diode efficiency $\eta$ calculated
from (b) and (c) using Eq. 1 (see text), as a function of $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$. (e–h) As in (a–d), but measured for
$V_{\mathrm{J}}=-1.5~{}\mathrm{V}$, which sets the switch JJ to the OFF state
and interrupts the right loop.

## Results and Discussion

### Flux-tunable multiterminal Josephson junction

The device under study, consisting of a multiterminal JJ embedded in a double-
loop geometry, is displayed in Fig. 1. It was realized in an InAs/Al
heterostructure [45, 46], where the epitaxial Al layer was selectively removed
to expose the III–V semiconductor below. We defined four superconducting
terminals, labeled S, L, M and R, coupled to a common semiconducting region.
Lithographically, the minimum distances between neighboring terminals were
$30~{}\mathrm{nm}$ (for L–M and R–M) and $50~{}\mathrm{nm}$ (for S–L and S–R),
while opposite terminals had separations of $100~{}\mathrm{nm}$ (L–R) and
$120~{}\mathrm{nm}$ (S–M). All junctions were short with respect to the
superconducting coherence length in the InAs 2D electron gas, estimated to be
approximately $600~{}\mathrm{nm}$ (see Methods section). Terminals L, M and R
were connected to a common node (D) forming two superconducting loops, which
enabled independent control over two phase differences [47, 48],
$\phi_{\mathrm{L}}-\phi_{\mathrm{M}}\equiv\phi_{\mathrm{L}}$ and
$\phi_{\mathrm{R}}-\phi_{\mathrm{M}}\equiv\phi_{\mathrm{R}}$ (here,
$\phi_{\alpha}$ indicates the superconducting phase of terminal
$\alpha\in\\{\mathrm{L,M,R}\\}$ and $\phi_{\mathrm{M}}$ was set to zero by
convention). This was achieved by passing currents $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$ through two flux-bias lines, patterned on top of a uniform
dielectric layer, resulting in external magnetic fluxes
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$ respectively
threading the left and right loop. Gate electrodes were deposited on the same
dielectric layer and energized by voltages $V_{\alpha}$
($\alpha\in\\{\mathrm{S,L,M,R}\\}$) and $V_{\mathrm{J}}$, allowing for
electrostatic tuning of the electron density in the InAs layer below. While
terminals L and M where directly connected to the node D via Al strips, a
planar JJ (named switch JJ) was integrated on terminal R. The switch JJ, with
a length of $40~{}\mathrm{nm}$ and a width of $5~{}\mathrm{\mu m}$, was
designed to have a critical current much larger than that between any pairs of
leads in the 4TJJ, and therefore the phase difference across the switch JJ can
be neglected for the following discussion. Depending on the gate voltage
$V_{\mathrm{J}}$, the switch JJ was employed in two configurations:
$V_{\mathrm{J}}=0$ (switch ON), where the JJ was conducting and
$\mathit{\Phi}_{\mathrm{R}}$ could be used to control $\phi_{\mathrm{R}}$, or
$V_{\mathrm{J}}=-1.5~{}\mathrm{V}$ (switch OFF), where the JJ was depleted,
the right loop was interrupted and terminal R was reduced to a floating
superconducting island. The other gate voltages were set to
$V_{\mathrm{S}}=0.1~{}\mathrm{V}$,
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$ and
$V_{\mathrm{M}}=-0.15~{}\mathrm{V}$, unless stated otherwise. The device was
measured in a dilution refrigerator with a base temperature of about
$10~{}\mathrm{mK}$. Current-bias experiments were performed in a four-terminal
configuration by driving a current $I_{\mathrm{SD}}$ between S and D and
measuring the voltage drop across the device, which allowed for the
measurement of the switching current $I_{\mathrm{sw}}$. Along its path between
S and D, the current flowed through the semiconducting region forming the
four-terminal JJ, and in particular across the S–L, S–M and S–R junctions. In
our geometry, the superconducting loops were designed to limit their
inductance, that could in principle lead to the SDE in the system. The maximum
flux variation due to the inductance of a loop, estimated to be approximately
$124~{}\mathrm{pH}$ (see details in the Supplementary Material, Section 5),
for a circulating current of the order of $100~{}\mathrm{nA}$, is $\sim
6\times 10^{-3}\mathit{\Phi}_{0}$. This was observed to be negligible with
respect to the flux scales over which the device properties varied. Further
details regarding materials, fabrication and measurement setup are provided in
the Methods section. Results on a second device, similar to the one discussed
in the Main Text, are presented in the Supplementary Material (see Figs.
S6–S10 in Section 4). Devices studied here were employed in a previous work
that investigated hybridization of ABSs in multiterminal JJs [47].

### Nonreciprocal supercurrents in the 2D phase space

First, we present the differential resistance $R$ as a function of the current
bias $I_{\mathrm{SD}}$ and of the left flux-line current $I_{\mathrm{L}}$ for
fixed right flux-line current $I_{\mathrm{R}}=0.1~{}\mathrm{mA}$. Here, $R$
was measured with standard lock-in techniques and $I_{\mathrm{SD}}$ was swept
from $0$ to positive or negative values to avoid retrapping effects. Figure 2a
shows the result for $V_{\mathrm{J}}=0$ (switch ON): the switching current
revealed oscillations of varying amplitude as a function of $I_{\mathrm{L}}$,
which, notably, were nonreciprocal at positive and negative $I_{\mathrm{SD}}$.
For instance, at $I_{\mathrm{L}}=0.18~{}\mathrm{mA}$ we measured switching
currents $I^{+}_{\mathrm{sw}}=58~{}\mathrm{nA}$ at $I_{\mathrm{SD}}>0$ and
$I^{-}_{\mathrm{sw}}=38~{}\mathrm{nA}$ at $I_{\mathrm{SD}}<0$ (see cyan
annotations), where $I^{\pm}_{\mathrm{sw}}>0$ by definition. This resulted in
a superconducting diode efficiency $\eta$, defined as:

$\eta=\frac{I^{+}_{\mathrm{sw}}-I^{-}_{\mathrm{sw}}}{I^{+}_{\mathrm{sw}}+I^{-}_{\mathrm{sw}}},$
(1)

of approximately $21\%$. We also note that the switching current vanished in a
small range around $I_{\mathrm{L}}\approx-60~{}\mathrm{\mu A}$ (yellow arrow),
namely the device had finite differential resistance at $I_{\mathrm{SD}}=0$.
Similar maps obtained at different settings of $I_{\mathrm{R}}$ are presented
in Fig. S1 of the Supplementary Material.

To efficiently measure the switching currents $I^{\pm}_{\mathrm{sw}}$ and the
diode efficiency $\eta$ as functions of both $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$, we changed measurement technique and periodically ramped
$I_{\mathrm{SD}}$ from zero to the amplitude $A=\pm 260~{}\mathrm{nA}$ with a
repetition rate of $133~{}\mathrm{Hz}$, and detected when the voltage drop
across the device exceeded a threshold. The time spent in the low-resistance
state, averaged over 32 consecutive measurements, was converted to a current,
resulting in a rapid measurement of $I^{+}_{\mathrm{sw}}$ or
$I^{-}_{\mathrm{sw}}$ (depending on the sign of $A$) displayed in Figs. 2b and
2c, respectively. A limitation of this measurement technique was that values
of $I_{\mathrm{sw}}$ below $\sim 10~{}\mathrm{nA}$ could not be accurately
detected due to the finite voltage threshold, which gave a finite reading of
about $10~{}\mathrm{nA}$ for small switching currents, and even when the
device was resistive for zero bias current. The switching current oscillated
periodically in the 2D phase space—where the periodicity axes correspond to
the external magnetic fluxes $\mathit{\Phi}_{\mathrm{L}}$ and
$\mathit{\Phi}_{\mathrm{R}}$ (black lines in Fig. 2b)—forming a pattern
characterized by lobe-like features. The finite slope of the
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$ axes with
respect to $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ was due to the cross-coupling
between the left (right) flux-bias line and the right (left) loop, as
discussed in Section 5 of the Supplementary Material. The oscillations of
$I_{\mathrm{sw}}$ exhibited maxima of approximately $250~{}\mathrm{nA}$ where
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$ were integer
multiples of the superconducting flux quantum $\mathit{\Phi}_{0}=h/2e$ (with
$h$ the Planck constant and $e$ the elementary charge), and minima at finite
phases where the limit of detection was reached, consistent with the vanishing
switching current discussed for Fig. 2a. We note that the switching currents
were nonreciprocal upon reversal of the current bias, while their 2D patterns
were symmetric to each other with respect to the origin
($I_{\mathrm{L}}=I_{\mathrm{R}}=0$, corresponding to
$\mathit{\Phi}_{\mathrm{L}}=\mathit{\Phi}_{\mathrm{R}}=0$). The symmetry was
particularly visible in the shape of the lobes, which was inverted as the
supercurrent changed sign. Figure 2d shows the superconducting diode
efficiency calculated from Figs. 2b and 2c using Eq. 1. As expected, $\eta$
reflected the 2D periodic pattern in the phase space of the switching
currents, and was widely tunable as a function of $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$. We observe a fully ambipolar character and large values up
to $\eta\approx\pm 21\%$ where $I^{\pm}_{\mathrm{sw}}$ had a large gradient in
the phase space, while the efficiency vanished in extended regions of the
phase space without the need for fine tuning $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$.

### Nonreciprocal supercurrents in single-loop configuration

Next, in Figs. 2e–h we present the measurements corresponding to those
discussed in Figs. 2a–d but with the switch junction in the OFF state
($V_{\mathrm{J}}=-1.5~{}\mathrm{V}$). From the differential resistance as a
function of $I_{\mathrm{SD}}$ and $I_{\mathrm{L}}$ (Fig. 2e, here for
$I_{\mathrm{R}}=17~{}\mathrm{\mu A}$), we found periodic oscillations of the
switching current, with $I^{+}_{\mathrm{sw}}$ and $I^{-}_{\mathrm{sw}}$
exhibiting a phase shift from each other and opposite skewness (in the forward
direction for $I^{+}_{\mathrm{sw}}$, backward for $I^{-}_{\mathrm{sw}}$).
Consequently, the switching currents were again nonreciprocal depending on
$I_{\mathrm{SD}}$, which indicates a large JDE; at
$I_{\mathrm{L}}=0.13~{}\mathrm{mA}$, for example,
$I^{+}_{\mathrm{sw}}=130~{}\mathrm{nA}$ and
$I^{-}_{\mathrm{sw}}=68~{}\mathrm{nA}$ (see cyan annotations), yielding
$\eta\approx 31\%$. We note that, in this configuration, the switching current
did not vanish for any value of $I_{\mathrm{L}}$, with minimal values of
$\approx 50~{}\mathrm{nA}$, unlike the case with the switch ON. Measurements
of $I^{+}_{\mathrm{sw}}$ and $I^{-}_{\mathrm{sw}}$ as functions of both
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ are shown in Figs. 2f and 2g. The 2D
pattern observed for $V_{\mathrm{J}}=0$ was no longer present: as expected,
the dependence on the flux $\mathit{\Phi}_{\mathrm{R}}$ was suppressed when
the right superconducting loop was interrupted, and periodicity remained along
a single direction. In agreement with Fig. 2e, the switching current
oscillations were shifted in phase and their skewness was reversed depending
on the sign of the current bias. The superconducting diode efficiency,
displayed in Fig. 2h for the data of panels f and g, was also characterized by
a periodic behavior as a function of $\mathit{\Phi}_{\mathrm{L}}$ and reached
maxima of approximately $34\%$.

Figure 3: Gate-tuning of the maximum switching current and diode efficiency.
(a) Differential resistance $R$ as a function of gate voltages
$V_{\mathrm{L}}=V_{\mathrm{R}}$ and bias current $I_{\mathrm{SD}}$ (swept from
negative to positive values, see white arrow). (b) Maximum Josephson diode
efficiency $\eta^{\mathrm{max}}$ as a function of the maximum switching
current $I_{\mathrm{sw}}^{\mathrm{max}}$ for multiple values of
$V_{\mathrm{L}}=V_{\mathrm{R}}$ (see colorscale). Each point is obtained from
maps similar to Figs. 2b and 2c (see Supplementary Material, Section 3 for
more details). Full circles refer to the situation with $V_{\mathrm{J}}=0$,
empty diamonds to the situation with $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$.
Figure 4: Routing of the supercurrent. (a) Switching current
$I^{+}_{\mathrm{sw}}$ measured for positive bias current $I_{\mathrm{SD}}$, as
a function of flux-line currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$.
Measurements are performed with $V_{\mathrm{L}}=-0.1~{}\mathrm{V}$ and
$V_{\mathrm{R}}=-0.5~{}\mathrm{V}$. (b) Diode efficiency $\eta$ for the
configuration of (a). (c, d) As in (a, b), but for
$V_{\mathrm{L}}=-0.5~{}\mathrm{V}$ and $V_{\mathrm{R}}=-0.1~{}\mathrm{V}$. (e,
f) As in (a, b), but for $V_{\mathrm{L}}=-0.1~{}\mathrm{V}$,
$V_{\mathrm{R}}=-0.1~{}\mathrm{V}$, and $V_{\mathrm{M}}=-1~{}\mathrm{V}$.

### Gate-tunable Josephson diode effect

Electrostatic tunability over the supercurrents and the JDE was enabled by
gates controlling the electron density in the semiconducting region of the
4TJJ. In Fig. 3a, we show the differential resistance as a function of the
current bias (here swept from negative to positive values, thus displaying
both retrapping and switching currents) while the gate voltages
$V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ varied simultaneously, for
$V_{\mathrm{J}}=0$ and $I_{\mathrm{L}}=I_{\mathrm{R}}=0$. The switching
current and the retrapping current, respectively $280~{}\mathrm{nA}$ and
$-265~{}\mathrm{nA}$ at $V_{\mathrm{L}}=V_{\mathrm{R}}=0$, decreased for more
negative voltages, until a finite resistance was measured at
$I_{\mathrm{SD}}=0$ for
$V_{\mathrm{L}}=V_{\mathrm{R}}\approx-0.35~{}\mathrm{V}$. To investigate the
voltage-tunability of the JDE, we measured the switching currents
$I^{\pm}_{\mathrm{sw}}$ as functions of $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$
(as in Figs. 2b, c, f, g) for varying $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$,
and, in each configuration, we extracted the peak values of
$I^{+}_{\mathrm{sw}}$, named $I_{\mathrm{sw}}^{\mathrm{max}}$, and of the
superconducting diode efficiency, $\eta^{\mathrm{max}}$ (see Supplementary
Material, Section 3 for the details of the extraction procedure). The result
is presented in Fig. 3b, where $\eta^{\mathrm{max}}$ is plotted as a function
of $I_{\mathrm{sw}}^{\mathrm{max}}$ for the two settings of the switch JJ
$V_{\mathrm{J}}=0$ and $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$. In both cases, we
observed an increasing trend of $\eta^{\mathrm{max}}$ as
$I_{\mathrm{sw}}^{\mathrm{max}}$ increased. For any gate setting, the diode
efficiency was larger at $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$ than at
$V_{\mathrm{J}}=0~{}\mathrm{V}$, up to $34\%$ and $21\%$ respectively (at
$V_{\mathrm{L}}=V_{\mathrm{R}}=0$), while $I_{\mathrm{sw}}^{\mathrm{max}}$ was
slightly smaller in the former case (up to $260~{}\mathrm{nA}$, instead of
$280~{}\mathrm{nA}$ for $V_{\mathrm{J}}=0$).

We further characterized the gate dependence of the device by allowing an
asymmetric tuning of $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ (at
$V_{\mathrm{J}}=0$), as shown in Figs. 4a, b and 4c, d for the configurations
$V_{\mathrm{L}}=-0.1~{}\mathrm{V},~{}V_{\mathrm{R}}=-0.5~{}\mathrm{V}$ and
$V_{\mathrm{L}}=-0.5~{}\mathrm{V},~{}V_{\mathrm{R}}=-0.1~{}\mathrm{V}$. In
each case, the switching current measured as a function of $I_{\mathrm{L}}$
and $I_{\mathrm{R}}$ for positive current bias is displayed in the first
panel, while the second panel presents the diode efficiency extracted from
$I^{+}_{\mathrm{sw}}$ and $I^{-}_{\mathrm{sw}}$. The two configurations
revealed a complementary behavior: the switching current oscillations and the
diode efficiency were almost completely suppressed as a function of
$\mathit{\Phi}_{\mathrm{L(R)}}$ when $V_{\mathrm{L(R)}}$ was set to
sufficiently negative value, depleting the semiconducting region between
terminals S and L (R). This highlights the possibility of routing the
supercurrents flowing in our device by gating, which enabled electrostatic
control over the phase dependence of the JDE. The results obtained for
$V_{\mathrm{R}}=-0.5~{}\mathrm{V}$ (panels a and b) were reminiscent of those
previously observed for $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$ (Figs. 2f–h), where
data were independent of $\mathit{\Phi}_{\mathrm{R}}$.

Finally, we restored the symmetric gate configuration
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$ and studied the effect of
depleting the middle gate $V_{\mathrm{M}}$, set to $-1~{}\mathrm{V}$ (see Fig.
4e and 4f). Here, we observed periodic oscillations of the switching current
along a single direction of the phase space, corresponding to the
$(\mathit{\Phi}_{\mathrm{L}}-\mathit{\Phi}_{\mathrm{R}})$-axis. The frequency
of these oscillations was doubled compared to the case where $V_{\mathrm{M}}$
was not depleted (e.g., Fig. 2b), consistent with the exclusion of terminal M
from the current path. As a consequence, screening currents induced by the
flux-bias lines only circulated in the perimeter of the double-loop geometry,
leading $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ to control the total flux
$\mathit{\Phi}_{\mathrm{L}}-\mathit{\Phi}_{\mathrm{R}}$ (note that
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$ were defined
with opposite signs in Figs. 1a and 1c). Notably, in this symmetric gate
configuration where no current flowed into terminal M, the JDE was essentially
suppressed (see Fig. 4f). Results for additional gate settings are shown in
the Supplementary Material, see Figs. S2–S5.

Figure 5: Minimal-model description of the Josephson diode effect. (a)
Schematic representation of the four-terminal Josephson junction and circuit
parameters. (b, c) Simulated critical currents $I^{+}_{\mathrm{c}}$ and
$I^{-}_{\mathrm{c}}$ for positive and negative $I_{\mathrm{SD}}$,
respectively, as functions of the phase differences $\phi_{\mathrm{L}}$ and
$\phi_{\mathrm{R}}$. The transmission of the three channels is $\tau=0.9$. (d)
Diode efficiency $\eta$ derived from (b) and (c) as a function of
$\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$. (e–h) Supercurrents in the four
leads as functions of phase $\phi_{\mathrm{S}}$. The phases of the other leads
are indicated in each panel. The four cases correspond to the colored markers
in (d). The value of $\phi_{\mathrm{S}}$ where $I_{\mathrm{S}}$ has its
maximum $I^{+}_{\mathrm{c}}$ (minimum $I^{-}_{\mathrm{c}}$), labeled
$\phi_{\mathrm{S}}^{+}$ ($\phi_{\mathrm{S}}^{-}$), is highlighted by the red
(green) dotted line. (i–l) Schematic representation of the supercurrent flow
in the phase configurations shown in (e–h). Red and green arrows indicate
supercurrents for $\phi_{\mathrm{S}}^{+}$ and $\phi_{\mathrm{S}}^{-}$,
respectively. The wider the arrow, the larger the supercurrent.

### Minimal-model description of JDE

To understand the behavior of our device in more depth, and the underlying
origin of the JDE, we introduce a simple circuit model that describes the
supercurrents of the 4TJJ. In the switch-ON configuration, terminals L, M and
R are all connected to the same node D, therefore the current can flow from S
to D (or from D to S) via three available JJs S–L, S–M and S–R. That is, the
4TJJ is mapped onto a bi-SQUID as three distinct JJs are connected in
parallel. The total current flowing into lead S is thus expressed as:

$I_{\mathrm{S}}=I_{\mathrm{SL}}+I_{\mathrm{SM}}+I_{\mathrm{SR}}=\sum_{\alpha}I_{\mathrm{S\alpha}},$
(2)

where $I_{\mathrm{S\alpha}}$ is the current flowing from terminal S to
terminal $\alpha$ via the corresponding JJ. First, we consider the minimal
model of a single numerical parameter, schematically shown in Fig. 5a. Each of
the three JJs, that are identical to each other, is described by one
conductive channel with transmission $\tau$, resulting in the CPR [49]:

$I_{\mathrm{S\alpha}}=\frac{e\mathit{\Delta}}{2\hbar}\frac{\tau\sin(\phi_{\mathrm{S}}-\phi_{\alpha})}{\sqrt{1-\tau\sin^{2}\frac{\phi_{\mathrm{S}}-\phi_{\alpha}}{2}}}.$
(3)

with $\mathit{\Delta}$ the superconducting gap
($\mathit{\Delta}=180~{}\mathrm{\mu eV}$ is used considering Al leads) and
$\hbar=h/2\pi$. We assume a high transmission $\tau=0.9$, for which the CPR of
Eq. 3 has a significantly nonsinusoidal character, as harmonics higher than
the first provide a sizable contribution. The independent variables in the
model are the three superconducting phase differences $\phi_{\mathrm{L}}$,
$\phi_{\mathrm{R}}$ and $\phi_{\mathrm{S}}$, defined with respect to
$\phi_{\mathrm{M}}\equiv 0$. The first two phases are related to the external
magnetic fluxes by
$\phi_{\mathrm{L(R)}}=2\pi\mathit{\Phi}_{\mathrm{L(R)}}/\mathit{\Phi}_{0}$
(neglecting the inductance of the loops, see discussion in the Supplementary
Material, Section 5), whereas $\phi_{\mathrm{S}}$ varies depending on current
bias $I_{\mathrm{SD}}$. The critical currents for the two bias directions are
then obtained as:

$I_{\mathrm{c}}^{\pm}=\max_{\phi_{\mathrm{S}}}\left[\pm
I_{\mathrm{S}}(\phi_{\mathrm{S}})\right].$ (4)

In Figs. 5b and 5c, we show $I^{\pm}_{\mathrm{c}}$ computed as functions of
$\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$, while the diode efficiency
calculated with Eq. 1 (where $I^{\pm}_{\mathrm{sw}}$ are substituted by
$I^{\pm}_{\mathrm{c}}$) is displayed in Fig. 5d. The critical currents,
fulfilling the condition
$I^{+}_{\mathrm{c}}(\phi_{\mathrm{L}},\phi_{\mathrm{R}})=I^{-}_{\mathrm{c}}(-\phi_{\mathrm{L}},-\phi_{\mathrm{R}})$,
exhibit patterns that are prominently asymmetric with respect to
$\phi_{\mathrm{L}}=\phi_{\mathrm{R}}=0$ (modulo $2\pi$), which leads to a
strong JDE with $\eta$ up to approximately $27\%$. The dependence of $\eta$ on
$\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$ reflects the triangular shapes
observed for $I^{\pm}_{\mathrm{c}}$, with features arranged according to three
main orientations in the phase space. The origin of the JDE is investigated by
fixing $\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$ and computing the CPR of
Eq. 2 as a function of $\phi_{\mathrm{S}}$,
$I_{\mathrm{S}}(\phi_{\mathrm{S}})$, and its components
$I_{\mathrm{SL}}(\phi_{\mathrm{S}})$, $I_{\mathrm{SM}}(\phi_{\mathrm{S}})$ and
$I_{\mathrm{SR}}(\phi_{\mathrm{S}})$, all obtained from Eq. 3. For simplicity
we always keep $\phi_{\mathrm{R}}=0$ and select four values of
$\phi_{\mathrm{L}}$ (colored markers in Fig. 5d), where $|\eta|$ is either
zero ($\phi_{\mathrm{L}}=0,\pi$) or maximal
($\phi_{\mathrm{L}}=0.78\pi,1.22\pi$). The individual and combined CPRs at
these phase-space points are plotted in Fig. 5e–h. In each case, we identify
the values of $\phi_{\mathrm{S}}$ that maximize the total current
$I_{\mathrm{S}}$ ($\phi_{\mathrm{S}}^{+}$, red dotted lines) and its inverse
$-I_{\mathrm{S}}$ ($\phi_{\mathrm{S}}^{-}$, green dotted lines), such that
$I_{\mathrm{S}}(\phi_{\mathrm{S}}^{\pm})=I^{\pm}_{\mathrm{c}}$. The currents
flowing to and from terminal S are schematically depicted in Fig. 5i–l for the
same $\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$ values of panels e–h. In the
schematics, red (green) arrows show the situation at
$\phi_{\mathrm{S}}^{+(-)}$, their width and direction indicate the magnitude
and direction of the current. We note that all individual CPRs
$I_{\mathrm{S\alpha}}(\phi_{\mathrm{S}})$ have the same amplitude $\approx
30~{}\mathrm{nA}$ and skewness, both given by the transmission $\tau$
(identical for the three channels), but notably
$I_{\mathrm{SL}}(\phi_{\mathrm{S}})$ is phase-shifted by $\phi_{\mathrm{L}}$.
When $\phi_{\mathrm{L}}=0$ (Fig. 5e, i), all components are in-phase and
$I_{\mathrm{S}}(\phi_{\mathrm{S}})=3I_{\mathrm{S\alpha}}(\phi_{\mathrm{S}})$,
hence a standard nonsinusoidal CPR is obtained. Positive and negative critical
currents are identical, and all currents are simply reversed between
$\phi_{\mathrm{S}}^{+}$ and $\phi_{\mathrm{S}}^{-}$. In contrast, when
$I_{\mathrm{SL}}(\phi_{\mathrm{S}})$ is shifted by $\phi_{\mathrm{L}}=0.78\pi$
(Fig. 5f, j), the total CPR becomes nonreciprocal for positive and negative
currents. The $I_{\mathrm{SL}}$-component is very small at
$\phi_{\mathrm{S}}^{+}$, but comparable with $I_{\mathrm{SM,SR}}$ at
$\phi_{\mathrm{S}}^{-}$; since in both cases $I_{\mathrm{SL}}$ has opposite
sign with respect to $I_{\mathrm{SM,SR}}$, this asymmetry leads to
$I^{+}_{\mathrm{c}}>I^{-}_{\mathrm{c}}$. A symmetric scenario is recovered for
$\phi_{\mathrm{L}}=\pi$, when $I_{\mathrm{SL}}$ is shifted by half period from
the other components. Here, $I_{\mathrm{S}}$ always has the opposite sign to
$I_{\mathrm{SM,SR}}$, but same absolute value at $\phi_{\mathrm{S}}^{\pm}$,
such that $I^{+}_{\mathrm{c}}=I^{-}_{\mathrm{c}}$ and no JDE is present.
Finally, the results obtained for $\phi_{\mathrm{L}}=1.22\pi$ (Fig. 5h,l),
symmetric about $\phi_{\mathrm{L}}=\pi$ to $\phi_{\mathrm{L}}=0.78\pi$, show
the same CPRs discussed for Fig. 5f,j upon sign inversion of both current and
$\phi_{\mathrm{S}}$, confirming that here the JDE is strong and has opposite
direction compared to the previous case.

We note that the JDE has been derived within our minimal model despite the
presence of three identical channels, i.e., of same harmonic content, whereas
in a conventional SQUID comprising two JJs the harmonic content must be
different between the two JJs [29]. The multiterminal geometry that we discuss
can also be reduced to a conventional SQUID, where two JJs and a phase degree
of freedom (for example, the S–M and S–R junctions and $\phi_{\mathrm{R}}$)
are replaced by an effective JJ of tunable harmonic content. The effective JJ,
together with the third JJ and the remaining phase difference (in the same
example, S–L and $\phi_{\mathrm{L}}$), realizes the proposal of Ref. [29]. An
important advantage offered by our platform lies in the possibility of phase-
tuning the harmonic content of the effective junction, establishing wide and
flexible control over the JDE. The nonsinusoidal character of the individual
CPRs, which is still a requirement, is obtained in high-transmission hybrid
JJs (as those realized in this work), while our novel geometry eliminates the
need for precise control over the transmissions of the single junctions after
device fabrication.

### Simulations with the extended model

Figure 6: Josephson diode effect in the extended model. (a) Schematic
representation of the four-terminal Josephson junction and circuit parameters
for the switch-ON configuration (see text for details). (b, c) Simulated
critical currents $I^{+}_{\mathrm{c}}$ and $I^{-}_{\mathrm{c}}$ for positive
and negative current bias (respectively), as functions of the flux-line
currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. Currents $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$ are calculated from the superconducting phase differences
$\phi_{\mathrm{L}}$, $\phi_{\mathrm{R}}$ and the mutual inductance matrix (see
discussion in the Supplementary Material, Section 5). Black arrows, whose
directions indicate the periodicity axes $\phi_{\mathrm{L}}$ and
$\phi_{\mathrm{R}}$, represent the corresponding phase winding by $2\pi$. (d)
Diode efficiency $\eta$ obtained from (b) and (c) as a function of
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. (e) Schematic representation of the
four-terminal Josephson junction and circuit parameters for the switch-OFF
configuration. (f, g) As in (b, c), but for the case with switch-OFF. (h)
Diode efficiency $\eta$ obtained from (f) and (g) as a function of
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$.

After discussing the origin of the JDE by means of a minimal, single-parameter
model, we extend the latter to better capture the experimental data presented
in Fig. 2. Again, we consider the three JJs S–L, S–M and S–R that contribute
to the total current according to Eq. 2. For each junction we consider two
contributions to the current $I_{\mathrm{S\alpha}}$: in addition to a high-
transparency channel, with transmission $\tau_{\alpha}$, a component with
conventional sinusoidal CPR [50], associated to a large number of low-
transmission channels, is included:

$I_{\mathrm{S\alpha}}=\frac{e\mathit{\Delta}}{2\hbar}\left[\frac{\tau_{\alpha}\sin(\phi_{\mathrm{S}}-\phi_{\alpha})}{\sqrt{1-\tau_{\alpha}\sin^{2}\frac{\phi_{\mathrm{S}}-\phi_{\alpha}}{2}}}+T_{\alpha}\sin(\phi_{\mathrm{S}}-\phi_{\alpha})\right],$
(5)

where $e\mathit{\Delta}T_{\alpha}/2\hbar$ is the critical current of the
sinusoidal component and $T_{\alpha}$ the sum of the transmissions of all low-
transmission channels. We note that $\tau_{\alpha}$ and $T_{\alpha}$ may vary
depending on the junction S–$\alpha$. With this extended model, that is
schematically represented in Fig. 6a, we compute the critical currents using
Eq. 4 for any settings of $\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$. The
simulated $I^{+}_{\mathrm{c}}$, $I^{-}_{\mathrm{c}}$ and $\eta$ are shown in
Fig. 6b–d for parameters $\tau_{\mathrm{L}}=\tau_{\mathrm{R}}=0.92$,
$\tau_{\mathrm{M}}=0.89$, $T_{\mathrm{L}}=3.5$, $T_{\mathrm{M}}=1.5$ and
$T_{\mathrm{R}}=3.6$. For a better comparison to the experimental results, the
three quantities are plotted as functions of the flux-bias line currents
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, calculated from the phases
$\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$ by applying a linear
transformation (see Supplementary Material, Section 5 for more details).
Simulations reproduced the measurements displayed in Fig. 2b–d to a good
degree. Calculated critical currents were between $10$ and
$270~{}\mathrm{nA}$, with diode efficiencies up to $\eta^{\mathrm{max}}\approx
25\%$ and patterns in the 2D phase space closely resembling the experimental
data. By comparing the simulation of $\eta$ in Fig. 6d to the result
previously obtained with the minimal model (Fig. 5d), we note a reduction of
$|\eta|$ and broadening of the features located near
$(\phi_{\mathrm{L}},\phi_{\mathrm{R}})=(\pi,\pi)$, modulo $2\pi$ (see black
arrows in both figures). This effect, also clearly visible in Fig. 2d, is
mainly related to the sinusoidal components to the CPR of Eq. 5, where
$T_{\mathrm{M}}$ is substantially smaller than $T_{\mathrm{L,R}}$, and
marginally related to $\tau_{\mathrm{M}}$, only slightly smaller than
$\tau_{\mathrm{L,R}}$. This is expected in the device under study, as the
larger length of the S–M JJ (lithographically of $120~{}\mathrm{nm}$) compared
to S–L and S–R ($50~{}\mathrm{nm}$) reduced both the transmission of the most
transmissive modes and the number of channels with low transmission.

The switch-OFF configuration (Fig. 2e–h) is investigated by further adjusting
the numerical model based on the following considerations. When the right
superconducting loop is interrupted, the current flowing from terminal S to R
does not have a direct path to D, but it must flow across the R–M junction.
Thus, we introduce this junction in the model, with CPR $I_{\mathrm{RM}}$ also
assumed to have the form of Eq. 5. Here the phase difference
$\phi_{\mathrm{R}}-\phi_{\mathrm{M}}=\phi_{\mathrm{R}}$ is used instead of
$\phi_{\mathrm{S}}-\phi_{\alpha}$ and parameters $\tau_{\mathrm{RM}}$ and
$T_{\mathrm{RM}}$ substitute $\tau_{\alpha}$ and $T_{\alpha}$ (see schematic
of Fig. 6e). For these parameters we choose the values
$\tau_{\mathrm{RM}}=0.97$ and $T_{\mathrm{RM}}=3.2$. The other consequence of
interrupting the right loop is that $\phi_{\mathrm{R}}$ is not controlled
externally with a magnetic flux, hence it is first calculated imposing the
condition
$I_{\mathrm{SR}}(\phi_{\mathrm{S}}-\phi_{\mathrm{R}})=I_{\mathrm{RM}}(\phi_{\mathrm{R}})$
(i.e., the current flowing from S to R equals that flowing from R to M), for
any value of $\phi_{\mathrm{S}}$. Once $\phi_{\mathrm{R}}$ is determined,
$I^{\pm}_{\mathrm{c}}(\phi_{\mathrm{L}})$ is computed using Eqs. 2, 4 and 5,
leading to the result shown in Figs. 6f and 6g. In agreement with Figs. 2f and
2g, the model produces oscillations of $I^{\pm}_{\mathrm{c}}$ as a function of
$\phi_{\mathrm{L}}$ between $50~{}\mathrm{nA}$ and $240~{}\mathrm{nA}$, with a
phase shift when reversing the current direction. This results in a diode
efficiency (see Fig. 6h) up to $40\%$, comparable to the measured value of
$\approx 34\%$ and exhibiting an oscillating behavior depending on
$\phi_{\mathrm{L}}$, similar to that in Fig. 2h.

The higher $\eta$ obtained in the switch-OFF case is understood by considering
the higher asymmetry in the supercurrent distribution obtained in this
setting. In fact, the supercurrent flowed directly from S to the common node D
only via S–L and S–M, which had largely different transmissions due to the
device geometry, while it had to traverse both S–R and R–M to reach D via R.
This realized a strongly asymmetric situation, where junctions with different
harmonic components led to large diode efficiencies [29]. When the switch was
ON (Figs. 2d and 6d) and junction S–R directly connected S to D, the
supercurrent distribution became more symmetric, and spatial-inversion
symmetry was broken by the different phases tuned with the loops: that is,
local magnetic fluxes broke both time-reversal and spatial-inversion symmetry.
Similar arguments apply to the case of Figs. 4b and 4d, however setting gates
to negative values to deplete parts of the semiconducting region reduced the
maximum switching current, which also resulted in a decrease of the JDE
efficiency (see Fig. 3). The absence of JDE for the situation of Fig. 4f is
explained by considering that, with terminal M blocked, the supercurrent
flowed in S–L and S–R only, which were almost balanced channels. Furthermore,
phase tuning could not break spatial-inversion symmetry with M blocked,
effectively resulting in a single superconducting loop between L and R. This
situation therefore realized the balanced SQUID device of Ref. [29], showing
no JDE despite the nonsinusoidal CPR of the individual junctions.

We finally note that our model captured the main experimental results well
without invoking hybridized ABSs in the semiconducting region, even though
such states were measured in the same devices by means of local tunneling
spectroscopy [47]. There, the terminal S was weakly coupled to the
multiterminal region and ABSs arising in the L–M and M–R junctions overlapped.
In the present case, these JJs were not required to describe the experimental
data (except for the switch-OFF configuration, where the M–R junction was not
phase-controllable), since the supercurrents flowing from S to L, M and R had
direct superconducting paths to D, without traversing L–M or M–R; this also
excluded the contribution of hybridized modes between the two JJs.
Furthermore, supercurrents were transported by several ABSs present in our
devices, most of which solely depended on phase differences between pairs of
superconducting contacts. Presumably, there were very few hybridized ABSs and
they were localized close to the center of the multiterminal region. Thus, in
the geometry studied here, they did not provide a significant contribution to
the total supercurrent.

## Conclusions

We reported switching current measurement of a 4TJJ in a InAs/Al
heterostructure hosting ABSs with large transmission probabilities, resulting
in nonsinusoidal CPRs between pairs of terminals. The switching current
measured between two contacts showed a strong dependence on the bias current
direction, resulting in a JDE. The JDE efficiency could be widely
controlled—both in amplitude and sign—by magnetic fluxes, independently tuned
via integrated flux-bias lines, and gate electrodes, which routed the
supercurrent to different transport channels. No external magnetic fields were
required. In a first gate setting, where transport through the entire
semiconductor region was allowed, the JDE efficiency was periodically
modulated by magnetic fluxes, with peak values reaching $\eta\approx\pm 21\%$,
including large regions in parameter space with $\eta\approx 0$. When a
superconducting arm was interrupted, introducing a larger asymmetry in the
supercurrent distribution, a peak efficiency $\eta\approx\pm 34\%$ was
reached. The 4TJJ was mapped onto a simple bi-SQUID geometry, with three
parallel JJs containing ABSs with large transmission probability. A
theoretical model reproduced the experimental observations to a good degree,
including switching current and diode efficiency patterns. In our devices, the
JDE is a consequence of the nonsinusoidal CPR and the multiterminal geometry,
which allows breaking of spatial-inversion symmetry by controlling the
magnetic fluxes in the loops. Unlike realizations based on a single loop, an
asymmetry between junctions is not required. Our work highlights the potential
of phase-tunable multiterminal JJs to engineer JDE with large and widely
controllable efficiencies, without the need for exotic materials or external
magnetic fields, and underscores the role of these devices as a versatile
platform for upcoming applications.

## Methods

### Materials and Fabrication

Devices were realized in a III–V heterostructure grown by molecular beam
epitaxy on an InP (001) substrate [46]. The semiconducting stack (starting
from the substrate) consisted of a 1100 nm thick step-graded InAlAs buffer
layer, a 6 nm thick $\mathrm{In_{0.75}Ga_{0.25}As}$ layer, an 8 nm thick InAs
layer, a 13 nm thick $\mathrm{In_{0.75}Ga_{0.25}As}$ layer and two monolayers
of GaAs. On top, an epitaxial 15 nm thick Al layer was deposited in situ
without breaking vacuum. A two-dimensional electron gas (2DEG) was confined in
the InAs and its properties were characterized via measurements performed in a
Hall bar geometry, which gave an electron peak mobility of
$18000~{}\mathrm{cm^{2}V^{-1}s^{-1}}$ at an electron sheet density of
${8\times 10^{11}~{}\mathrm{cm^{-2}}}$. This resulted in an electron mean free
path $l_{e}\gtrsim 260~{}\mathrm{nm}$ and a superconducting coherence length
of the 2DEG proximitized by the Al sheet of $\xi_{\mathrm{InAs}}=\sqrt{\hbar
v_{\mathrm{F}}l_{e}/\left(\pi\mathit{\Delta}\right)}\sim 600~{}\mathrm{nm}$,
with $v_{\mathrm{F}}$ the Fermi velocity in the 2DEG and $\mathit{\Delta}$ the
induced superconducting gap.

In the fabrication process, large mesa structures were first isolated,
suppressing parallel conduction between devices and across the middle regions
of the superconducting loops. This was done by selectively etching the Al
layer with Transene type D, followed by a second chemical etch to a depth of
$\sim 380~{}\mathrm{nm}$ into the III–V material stack, using a $220:55:3:3$
solution of $\mathrm{H_{2}O:C_{6}H_{8}O_{7}:H_{3}PO_{4}:H_{2}O_{2}}$. Next,
features were defined in the Al layer by wet etching with Transene type D at
$50\mathrm{{}^{\circ}C}$ for $4~{}\mathrm{s}$. The dielectric, uniformly
deposited on the entire chip by atomic layer deposition, consisted of a
$3~{}\mathrm{nm}$ thick layer of $\mathrm{Al_{2}O_{3}}$ and a
$15~{}\mathrm{nm}$ thick layer of $\mathrm{HfO_{2}}$. Gate electrodes and
flux-bias lines were defined by evaporation and lift-off. In a first step,
$5~{}\mathrm{nm}$ of Ti and $20~{}\mathrm{nm}$ of Au were deposited to realize
the fine features of the gates; in a second step, a stack of Ti/Al/Ti/Au with
thicknesses $5~{}\mathrm{nm}$, $340~{}\mathrm{nm}$, $5~{}\mathrm{nm}$ and
$100~{}\mathrm{nm}$ was deposited to connect the mesa structure to the bonding
pads and to define the flux-bias lines.

### Measurements Techniques

Experiments were performed in a dilution refrigerator with a base temperature
at the mixing chamber below $10~{}\mathrm{mK}$. The sample was mounted on a
QDevil QBoard sample holder system, without employing any light-tight
enclosure. Electrical contacts to the devices, excepts for the flux-bias
lines, were provided via a resistive loom with QDevil RF and RC low-pass
filters at the mixing chamber stage, and RC low-pass filters integrated on the
QBoard sample holder. Currents were passed through the flux-bias lines via a
superconducting loom with only QDevil RF filters at the mixing chamber stage.
Signals were applied to all gates and flux-bias lines via home-made RC filters
at room temperature.

In all electrical measurements, a bias current $I_{\mathrm{SD}}$ was driven
between terminal S and node D of the device by applying equal and opposite
voltages to S and D via bias resistors, whose resistance was much larger than
that of the device under study. The voltage drop across S and D was detected
in a four-terminal configuration. Measurements of the differential resistance
were performed with lock-in-amplifier techniques, by applying a fixed AC
current $\delta I=2.5~{}\mathrm{nA}$ to D in addition to the DC bias
$I_{\mathrm{SD}}$ and detecting the AC voltage $\delta V$ between S and D,
thus obtaining the differential resistance $R\equiv\delta V/\delta I$.
Measurements of $I^{\pm}_{\mathrm{sw}}$ were done by periodically ramping
$I_{\mathrm{SD}}$ from $0$ to an amplitude $A$, where $A$ was positive or
negative depending on whether $I^{+}_{\mathrm{sw}}$ or $I^{-}_{\mathrm{sw}}$
was measured; the absolute value of $A$ was adjusted depending on the gate
configuration to be slightly larger than $I^{\pm}_{\mathrm{sw}}$. The signal
form was a sawtooth wave, applied at a frequency $f=133~{}\mathrm{Hz}$ using a
waveform generator. The voltage drop across S and D was measured with an
oscilloscope (averaging 32 measurements), which detected the time interval
$\Delta t$ where the voltage was below a threshold, hence allowing for the
calculation of the switching current as $I^{\pm}_{\mathrm{sw}}=|A|f\Delta t$.

The dilution refrigerator was equipped with a superconducting vector magnet
which, despite not being utilized for the experiments, produced a small
magnetic field offset. Therefore, small offsets in the flux-bias line currents
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ (up to $\sim 100~{}\mathrm{\mu A}$) were
considered in datasets, in such a manner that the point where
$I_{\mathrm{L}}=I_{\mathrm{R}}=0$ corresponded to a point of the phase space
where $\eta=0$ and $I^{\pm}_{\mathrm{sw}}$ were maximal, as expected when no
magnetic fluxes thread the superconducting loops.

## Data availability

Data presented in this work will be available on Zenodo. The data that support
the findings of this study are available upon request from the corresponding
author.

## Acknowledgments

We acknowledge fruitful discussions with R. Seoane Souto and H. Weisbrich. We
thank the Cleanroom Operations Team of the Binnig and Rohrer Nanotechnology
Center (BRNC) for their help and support. W.W. acknowledges support from the
Swiss National Science Foundation (grant number 200020_207538). J.C.C.
acknowledges support from the Spanish Ministry of Science and Innovation
(Grant No. PID2020-114880GB-I00) and thanks the Deutsche
Forschungsgemeinschaft (DFG; German Research Foundation) via SFB 1432 for
sponsoring his stay at the University of Konstanz as a Mercator Fellow. W.B.
acknowledges support from the DFG via SFB 1432 (ID 425217212) and BE 3803/14-1
(ID 467596333). F.N. acknowledges support from the European Research Council
(grant number 804273) and the Swiss National Science Foundation (grant number
200021_201082).

## References

  * Sze and Ng [2007] S. M. Sze and K. K. Ng,  _Physics of Semiconductor Devices_ , 3rd ed. (Wiley, 2007).
  * Nadeem  _et al._ [2023] M. Nadeem, M. S. Fuhrer, and X. Wang, Nat. Rev. Phys. 5, 558 (2023).
  * Fulton and Dynes [1970] T. A. Fulton and R. C. Dynes, Phys. Rev. Lett. 25, 794 (1970).
  * Fulton  _et al._ [1972] T. A. Fulton, L. N. Dunkleberger, and R. C. Dynes, Phys. Rev. B 6, 855 (1972).
  * Tsang and Duzer [1975] W.-T. Tsang and T. V. Duzer, J. Appl. Phys. 46, 4573 (1975).
  * Sivakov  _et al._ [2018] A. G. Sivakov, O. G. Turutanov, A. E. Kolinko, and A. S. Pokhila, Low Temp. Phys. 44, 226 (2018).
  * Ando  _et al._ [2020] F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Nature 584, 373 (2020).
  * Lyu  _et al._ [2021] Y.-Y. Lyu, J. Jiang, Y.-L. Wang, Z.-L. Xiao, S. Dong, Q.-H. Chen, M. V. Milošević, H. Wang, R. Divan, J. E. Pearson, P. Wu, F. M. Peeters, and W.-K. Kwok, Nat. Commun. 12, 2703 (2021).
  * Daido  _et al._ [2022] A. Daido, Y. Ikeda, and Y. Yanase, Phys. Rev. Lett. 128, 037001 (2022).
  * Ilić and Bergeret [2022] S. Ilić and F. S. Bergeret, Phys. Rev. Lett. 128, 177001 (2022).
  * He  _et al._ [2022] J. J. He, Y. Tanaka, and N. Nagaosa, New J. Phys. 24, 053014 (2022).
  * Bauriedl  _et al._ [2022] L. Bauriedl, C. Bäuml, L. Fuchs, C. Baumgartner, N. Paulik, J. M. Bauer, K.-Q. Lin, J. M. Lupton, T. Taniguchi, K. Watanabe, C. Strunk, and N. Paradiso, Nat. Commun. 13, 4266 (2022).
  * Hou  _et al._ [2023] Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ilić, O. Glezakou-Elbert, A. Varambally, F. S. Bergeret, A. Kamra, L. Fu, P. A. Lee, and J. S. Moodera, Phys. Rev. Lett. 131, 027001 (2023).
  * Baumgartner  _et al._ [2022a] C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Nat. Nanotechnol. 17, 39 (2022a).
  * Baumgartner  _et al._ [2022b] C. Baumgartner, L. Fuchs, A. Costa, J. Picó-Cortés, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. F. Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, J. Phys.: Condens. Matter 34, 154005 (2022b).
  * Costa  _et al._ [2023] A. Costa, C. Baumgartner, S. Reinhardt, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, J. Fabian, D. Kochan, N. Paradiso, and C. Strunk, Nat. Nanotechnol. 18, 1266 (2023).
  * Lotfizadeh  _et al._ [2023] N. Lotfizadeh, B. Pekerten, P. Yu, W. Strickland, A. Matos-Abiague, and J. Shabani, Superconducting diode effect sign change in epitaxial Al-InAs Josepshon junctions, arXiv:2303.01902 (2023).
  * Davydova  _et al._ [2022] M. Davydova, S. Prembabu, and L. Fu, Sci. Adv. 8, 23 (2022).
  * Pal  _et al._ [2022] B. Pal, A. Chakraborty, P. K. Sivakumar, M. Davydova, A. K. Gopi, A. K. Pandeya, J. A. Krieger, Y. Zhang, M. Date, S. Ju, N. Yuan, N. B. M. Schröter, L. Fu, and S. S. P. Parkin, Nat. Phys. 18, 1228 (2022).
  * Yuan and Fu [2022] N. F. Q. Yuan and L. Fu, Proc. Natl. Acad. Sci. U.S.A. 119, e2119548119 (2022).
  * Wu  _et al._ [2022] H. Wu, Y. Wang, Y. Xu, P. K. Sivakumar, C. Pasco, U. Filippozzi, S. S. P. Parkin, Y.-J. Zeng, T. McQueen, and M. N. Ali, Nature 604, 653 (2022).
  * Narita  _et al._ [2022] H. Narita, J. Ishizuka, R. Kawarazaki, D. Kan, Y. Shiota, T. Moriyama, Y. Shimakawa, A. V. Ognev, A. S. Samardak, Y. Yanase, and T. Ono, Nat. Nanotechnol. 17, 823 (2022).
  * Jeon  _et al._ [2022] K.-R. Jeon, J.-K. Kim, J. Yoon, J.-C. Jeon, H. Han, A. Cottet, T. Kontos, and S. S. P. Parkin, Nat. Mater. 21, 1008 (2022).
  * Lin  _et al._ [2022] J.-X. Lin, P. Siriviboon, H. D. Scammell, S. Liu, D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, M. S. Scheurer, and J. Li, Nat. Phys. 18, 1221 (2022).
  * Zhang  _et al._ [2022] Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, Phys. Rev. X 12, 041013 (2022).
  * Kokkeler  _et al._ [2022] T. H. Kokkeler, A. A. Golubov, and F. S. Bergeret, Phys. Rev. B 106, 214504 (2022).
  * Díez-Mérida  _et al._ [2023] J. Díez-Mérida, A. Díez-Carlón, S. Y. Yang, Y.-M. Xie, X.-J. Gao, J. Senior, K. Watanabe, T. Taniguchi, X. Lu, A. P. Higginbotham, K. T. Law, and D. K. Efetov, Nat. Commun. 14, 2396 (2023).
  * Yun  _et al._ [2023] J. Yun, S. Son, J. Shin, G. Park, K. Zhang, Y. J. Shin, J.-G. Park, and D. Kim, Phys. Rev. Res. 5, L022064 (2023).
  * Souto  _et al._ [2022] R. S. Souto, M. Leijnse, and C. Schrade, Phys. Rev. Lett. 129, 267702 (2022).
  * Fominov and Mikhailov [2022] Y. V. Fominov and D. S. Mikhailov, Phys. Rev. B 106, 134514 (2022).
  * English  _et al._ [2016] C. D. English, D. R. Hamilton, C. Chialvo, I. C. Moraru, N. Mason, and D. J. Van Harlingen, Phys. Rev. B 94, 115435 (2016).
  * Nichele  _et al._ [2020] F. Nichele, E. Portolés, A. Fornieri, A. M. Whiticar, A. C. C. Drachmann, S. Gronin, T. Wang, G. C. Gardner, C. Thomas, A. T. Hatke, M. J. Manfra, and C. M. Marcus, Phys. Rev. Lett. 124, 226801 (2020).
  * Aggarwal  _et al._ [2021] K. Aggarwal, A. Hofmann, D. Jirovec, I. Prieto, A. Sammak, M. Botifoll, S. Martí-Sánchez, M. Veldhorst, J. Arbiol, G. Scappucci, J. Danon, and G. Katsaros, Phys. Rev. Res. 3, L022005 (2021).
  * Haxell  _et al._ [2023] D. Z. Haxell, M. Coraiola, M. Hinderling, S. C. ten Kate, D. Sabonis, A. E. Svetogorov, W. Belzig, E. Cheah, F. Krizek, R. Schott, W. Wegscheider, and F. Nichele, Nano Lett. 23, 7532 (2023).
  * Ciaccia  _et al._ [2023a] C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Lindemann, M. J. Manfra, C. Schrade, and C. Schönenberger, Phys. Rev. Res. 5, 033131 (2023a).
  * Ciaccia  _et al._ [2023b] C. Ciaccia, R. Haller, A. C. C. Drachmann, T. Lindemann, M. J. Manfra, C. Schrade, and C. Schönenberger, Charge-4e supercurrent in an InAs-Al superconductor-semiconductor heterostructure, arXiv:2306.07109 (2023b).
  * Valentini  _et al._ [2023] M. Valentini, O. Sagi, L. Baghumyan, T. de Gijsel, J. Jung, S. Calcaterra, A. Ballabio, J. A. Servin, K. Aggarwal, M. Janik, T. Adletzberger, R. S. Souto, M. Leijnse, J. Danon, C. Schrade, E. Bakkers, D. Chrastina, G. Isella, and G. Katsaros, Radio frequency driven superconducting diode and parity conserving Cooper pair transport in a two-dimensional germanium hole gas, arXiv:2306.07109 (2023).
  * Gupta  _et al._ [2023] M. Gupta, G. V. Graziano, M. Pendharkar, J. T. Dong, C. P. Dempsey, C. Palmstrøm, and V. S. Pribiag, Nat. Commun. 14, 3078 (2023).
  * Chiles  _et al._ [2023] J. Chiles, E. G. Arnault, C.-C. Chen, T. F. Q. Larson, L. Zhao, K. Watanabe, T. Taniguchi, F. Amet, and G. Finkelstein, Nano Lett. 23, 5257 (2023).
  * Zhang  _et al._ [2023] F. Zhang, A. S. Rashid, M. T. Ahari, G. J. de Coster, T. Taniguchi, K. Watanabe, M. J. Gilbert, N. Samarth, and M. Kayyalha, Magnetic-field-free nonreciprocal transport in graphene multi-terminal Josephson junctions, arXiv:2301.05081 (2023).
  * Pillet  _et al._ [2023] J.-D. Pillet, S. Annabi, A. Peugeot, H. Riechert, E. Arrighi, J. Griesmar, and L. Bretheau, Phys. Rev. Res. 5, 033199 (2023).
  * Matsuo  _et al._ [2023] S. Matsuo, T. Imoto, T. Yokoyama, Y. Sato, T. Lindemann, S. Gronin, G. C. Gardner, M. J. Manfra, and S. Tarucha, Nat. Phys. 19, 1636 (2023).
  * Leroux  _et al._ [2022] C. Leroux, A. Parra-Rodriguez, R. Shillito, A. D. Paolo, W. D. Oliver, C. M. Marcus, M. Kjaergaard, A. Gyenis, and A. Blais, Nonreciprocal devices based on voltage-tunable junctions, arXiv:2209.06194 (2022).
  * Virtanen and Heikkilä [2023] P. Virtanen and T. T. Heikkilä, Nonreciprocal Josephson linear response, arXiv:2306.12295 (2023).
  * Shabani  _et al._ [2016] J. Shabani, M. Kjaergaard, H. J. Suominen, Y. Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’l, S. Kraemer, C. Nayak, M. Troyer, C. M. Marcus, and C. J. Palmstrøm, Phys. Rev. B 93, 155402 (2016).
  * Cheah  _et al._ [2023] E. Cheah, D. Z. Haxell, R. Schott, P. Zeng, E. Paysen, S. C. ten Kate, M. Coraiola, M. Landstetter, A. B. Zadeh, A. Trampert, M. Sousa, H. Riel, F. Nichele, W. Wegscheider, and F. Krizek, Phys. Rev. Mater. 7, 073403 (2023).
  * Coraiola  _et al._ [2023a] M. Coraiola, D. Z. Haxell, D. Sabonis, H. Weisbrich, A. E. Svetogorov, M. Hinderling, S. C. ten Kate, E. Cheah, F. Krizek, R. Schott, W. Wegscheider, J. C. Cuevas, W. Belzig, and F. Nichele, Nat. Commun. 14, 6784 (2023a).
  * Coraiola  _et al._ [2023b] M. Coraiola, D. Z. Haxell, D. Sabonis, M. Hinderling, S. C. t. Kate, E. Cheah, F. Krizek, R. Schott, W. Wegscheider, and F. Nichele, Spin-degeneracy breaking and parity transitions in three-terminal josephson junctions, arXiv:2307.06715 (2023b).
  * Beenakker [1991] C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991).
  * Tinkham [2004] M. Tinkham,  _Introduction to Superconductivity_ , 2nd ed. (Dover Publications, 2004).
  * [51] Geometric inductance $L_{\mathrm{geom}}$ of a loop was calculated analytically using the following expression, valid for a rectangular loop: $L_{\mathrm{geom}}=\frac{\mu_{0}}{\pi}\Bigr{[}-2(W+H)+2\sqrt{H^{2}+W^{2}}-H\ln\left(\frac{H+\sqrt{H^{2}+W^{2}}}{W}\right)-W\ln\left(\frac{W+\sqrt{H^{2}+W^{2}}}{H}\right)+H\ln\left(\frac{4H}{d}\right)+W\ln\left(\frac{4W}{d}\right)\Bigr{]}$. Here, $H=26~{}\mathrm{\mu m}$ and $W=4.5~{}\mathrm{\mu m}$ are the dimensions of the loop, $d=1~{}\mathrm{\mu m}$ is the width of the Al strip forming the loop and $\mu_{0}$ is the vacuum magnetic permeability.
  * Annunziata  _et al._ [2010] A. J. Annunziata, D. F. Santavicca, L. Frunzio, G. Catelani, M. J. Rooks, A. Frydman, and D. E. Prober, Nanotechnology 21, 445202 (2010).

## Supplementary Material

##  1 Additional Phase-Space Linecuts

The switching current measured for positive bias current $I^{+}_{\mathrm{sw}}$
in the phase space (i.e., as a function of both flux-line currents
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$), shown in Fig. 2b of the Main Text, is
plotted again in Fig. S.1a over extended ranges of $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$. Here, in addition to the phase-space linecut
$I_{\mathrm{R}}=100~{}\mathrm{\mu A}$ displayed in the Main Text (green
marker), we consider the linecuts $I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ (white
marker) and $I_{\mathrm{R}}=-60~{}\mathrm{\mu A}$ (blue marker). Along these
directions, we measured the differential resistance $R$ across the device as a
function of $I_{\mathrm{L}}$ and of the bias current $I_{\mathrm{SD}}$ (always
swept from $0$ to positive or negative values), and show the result in Figs.
S.1b–d. The three linecuts revealed oscillations of the switching current as a
function of $I_{\mathrm{L}}$, with maximum switching currents of approximately
$240~{}\mathrm{nA}$. In all measurements, the supercurrent was nonreciprocal
at positive and negative $I_{\mathrm{SD}}$, confirming the presence of
Josephson diode effect (JDE) with phase-dependent efficiency $\eta$,
consistent with Fig. 2d of the Main Text.

##  2 Results for Different $V_{\mathrm{L}}$, $V_{\mathrm{R}}$

Figures S.2a–c show three phase-space linecuts, indicated by the colored
markers in Fig. S.1a, where the gate voltages $V_{\mathrm{L}}$ and
$V_{\mathrm{R}}$ were set to $-0.2~{}\mathrm{V}$ (while
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$ in Fig. 2 of the Main Text
and in Fig. S.1). The other gate voltages were kept to
$V_{\mathrm{S}}=0.1~{}\mathrm{V}$, $V_{\mathrm{M}}=-0.15~{}\mathrm{V}$ and
$V_{\mathrm{J}}=0$. The switching current oscillations qualitatively resembled
those described for $V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$ (Fig.
S.1) but with reduced amplitude, as the maximum switching current was
approximately $170~{}\mathrm{nA}$ in agreement with Fig. 3b of the Main Text.
Regions of vanishing supercurrent (see for example the yellow arrow in Fig.
S.2a) were more prominent in this gate configuration, consistent with the
larger normal-state resistance. The linecut of Fig. S.2b
($I_{\mathrm{R}}=20~{}\mathrm{\mu A}$) was also measured with the switch
junction voltage set to $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$ (switch OFF, Fig.
S.2d), yielding a picture very similar to Fig. 2e of the Main Text. With
respect to the case where $V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$,
we confirm a reduction of the maximum supercurrent (up to about
$170~{}\mathrm{nA}$) and diode efficiency (up to approximately $25\%$).

The results shown thus far were acquired with symmetric gate voltages
$V_{\mathrm{L}}=V_{\mathrm{R}}$, except in Fig. 4a–d of the Main Text where
either gate was strongly depleted. In these configurations, the currents
flowing into terminals L and R were relatively symmetric, as supported by our
simulations (Fig. 6), where a good fit was found for
$\tau_{\mathrm{L}}=\tau_{\mathrm{R}}$ and $T_{\mathrm{L}}\approx
T_{\mathrm{R}}$. For $V_{\mathrm{L}}=V_{\mathrm{R}}=-0.3~{}\mathrm{V}$, we
observed switching current patterns that were slightly asymmetric along the
two periodicity directions, as shown in Figs. S.3a and S.3b (see cyan arrows
in panel a). This was likely due to different lever arm of the gates at
voltages $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$, which created an imbalance in
the device for intermediate gate voltages. By setting
$V_{\mathrm{L}}=-0.28~{}\mathrm{V}$ and $V_{\mathrm{R}}=-0.31~{}\mathrm{V}$, a
more balanced situation was restored (see Figs. S.3d and S.3e). This effect
was also visible in the superconducting diode efficiencies, shown in Figs.
S.3c and S.3f for the two cases: while the maximum $|\eta|$ slightly varied
depending on the periodicity axis at
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.3~{}\mathrm{V}$, it was substantially more
symmetric in the two directions at $V_{\mathrm{L}}=-0.28~{}\mathrm{V}$,
$V_{\mathrm{R}}=-0.31~{}\mathrm{V}$ (see black arrows in panels c and f,
respectively).

Staying in the gate configuration with $V_{\mathrm{L}}=-0.28~{}\mathrm{V}$,
$V_{\mathrm{R}}=-0.31~{}\mathrm{V}$, we defined two phase-space linecuts
(colored markers in Fig. S.3d, corresponding to
$I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ and $I_{\mathrm{R}}=-140~{}\mathrm{\mu
A}$) and measured $R$ along these directions as a function of $I_{\mathrm{L}}$
and $I_{\mathrm{SD}}$. The result is presented in Figs. S.4a and S.4b, showing
switching current oscillations with maxima of about $70~{}\mathrm{nA}$ and
minima where the supercurrent vanished. Nonreciprocity was still present in
the switching current depending on the sign of $I_{\mathrm{SD}}$, despite less
markedly than for the previous gate configurations (where $V_{\mathrm{L}}$ and
$V_{\mathrm{R}}$ were set to less negative values), consistent with diode
efficiencies up to approximately $10\%$ reported in Fig. S.3f. Finally, the
gate voltage $V_{\mathrm{J}}$ was set to $-1.5~{}\mathrm{V}$ to operate with
the switch OFF, and supercurrent oscillations were measured along the linecut
at $I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ (see Fig. S.4c), also revealing
maximum switching currents of close to $70~{}\mathrm{nA}$.

##  3 Extracting the Diode Efficiency as a Function of $V_{\mathrm{L}}$,
$V_{\mathrm{R}}$

In Fig. 3b of the Main Text, we showed the gate-tunability of the JDE in our
devices by presenting the dependence of the maximum diode efficiency
$\eta^{\mathrm{max}}$ as a function of the gate voltages
$V_{\mathrm{L}}=V_{\mathrm{R}}$. In the switch-ON configuration
($V_{\mathrm{J}}=0$), each data point was extracted from measurements of
$I^{+}_{\mathrm{sw}}$ and $I^{-}_{\mathrm{sw}}$ taken as a function of both
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. Figures S.5a–e and S.5f–j display
$I^{+}_{\mathrm{sw}}$ and $\eta$ obtained using Eq. 1 of the Main Text for
five selected gate configurations ranging between
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.3~{}\mathrm{V}$ and
$V_{\mathrm{L}}=V_{\mathrm{R}}=0$. Since values close to the limit of
detection ($\sim 10~{}\mathrm{nA}$) gave large variability of the extracted
$\eta$ (see for example Figs. S.3c and S.3f in the regions of phase space
where $I^{+}_{\mathrm{sw}}$ was $\gtrsim 10~{}\mathrm{nA}$), regions where
$I^{+}_{\mathrm{sw}}$ was lower than a threshold of $20~{}\mathrm{nA}$ were
not considered. We note that the exact choice of the threshold did not
significantly alter the result of the extraction. In each gate configuration,
$I_{\mathrm{sw}}^{\mathrm{max}}$ and $\eta^{\mathrm{max}}$ were obtained by
considering the 99.9th percentile of $I^{+}_{\mathrm{sw}}$ and $\eta$,
respectively, throughout the corresponding maps. In switch-OFF case
($V_{\mathrm{J}}=-1.5~{}\mathrm{V}$), included in Fig. 3b of the Main Text,
the extraction was simplified for two reasons: first, since only one
periodicity axis remained in the phase space, $I^{\pm}_{\mathrm{sw}}$ could be
measured along a single direction (e.g., as a function of $I_{\mathrm{L}}$ for
any fixed $I_{\mathrm{R}}$), thus improving both speed and resolution of the
measurement; second, $I^{\pm}_{\mathrm{sw}}$ did not approach the limit of
detection very closely, therefore a threshold was not required to select the
data. Similar to the switch-ON case, $I_{\mathrm{sw}}^{\mathrm{max}}$ and
$\eta^{\mathrm{max}}$ were obtained by considering the 99.9 percentile of
$I^{+}_{\mathrm{sw}}$ and $\eta$ for each gate setting.

##  4 Results for Device 2

To further study the JDE arising in a four-terminal Josephson junction (4TJJ)
embedded in a double-loop geometry, we characterized a second device
fabricated on the same chip. The circuit layout of Device 2 and the geometry
of the two superconducting loops were lithographically identical to Device 1
(see Figs. 1a and 1c of the Main Text). The 4TJJ region, displayed in Fig.
S.6, featured a different layout of terminal S (significantly wider than for
Device 1), whereas terminals L, M and R were designed to be identical between
the two devices. The shape of the gates was also varied, in particular for the
gate energized by the voltage $V_{\mathrm{S}}$.

Current-bias measurements were performed as described for Device 1. Figure S.7
shows the switching current $I^{\pm}_{\mathrm{sw}}$ and the extracted diode
efficiency $\eta$ as functions of the flux-line currents $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$ for three configurations of the gate voltages
$V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ (set to a common value):
$-0.4~{}\mathrm{V}$ (a–c), $-0.5~{}\mathrm{V}$ (d–f) and $-0.6~{}\mathrm{V}$
(g–i). The other gate voltages were $V_{\mathrm{S}}=0.15~{}\mathrm{V}$,
$V_{\mathrm{M}}=-0.15~{}\mathrm{V}$ and $V_{\mathrm{J}}=0$ (switch ON).
Switching current oscillations in the 2D phase space at
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.4~{}\mathrm{V}$ qualitatively resemble those
observed for Device 1 in Figs. 2b and 2c of the Main Text. The larger current
maxima and minima, occurring despite voltages were applied in a more negative
range than in Device 1, are understood by considering the different geometry
of terminal S, that is compatible with a higher number of conduction channels
in the S–L and S–R junctions, and a stronger screening of the electric field
generated by the gates. Another feature of Device 2, that was particularly
visible at
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.5~{}\mathrm{V},~{}-0.6~{}\mathrm{V}$, was
the large asymmetry between the two periodicity directions
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$. This is
attributed to a smaller supercurrent flowing from S to L than from S to R,
likely due to a combination of a smaller number and transmission of the modes
on the left side of the device, even for symmetrically applied gate voltages.
We note that the case with $V_{\mathrm{L}}=V_{\mathrm{R}}=-0.6~{}\mathrm{V}$
is qualitatively similar to Figs. 4c and 4d of the Main Text, where only
$V_{\mathrm{L}}$ was depleted. The asymmetry was reflected in the diode
efficiency $\eta$: while oscillations of $\eta$ in the phase space had some
resemblance with those of Device 1 (see Figs. 2d of the Main Text), and also
reached large maxima up to $|\eta|\approx 28\%$, they showed different
amplitude and features depending on which flux was varied. A prominent example
is visible in Fig. S.7c, where features of $\eta$ near the phase-space points
$(\mathit{\Phi}_{\mathrm{L}},\mathit{\Phi}_{\mathrm{R}})=(0,\mathit{\Phi}_{0}/2)$
(modulo $\mathit{\Phi}_{0}$), marked by the orange arrow, have different shape
and smaller $\eta$ than those near
$(\mathit{\Phi}_{\mathrm{L}},\mathit{\Phi}_{\mathrm{R}})=(\mathit{\Phi}_{0}/2,0)$
(purple arrow). Conversely, for
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.5~{}\mathrm{V},~{}-0.6~{}\mathrm{V}$, $\eta$
was larger in proximity
$(\mathit{\Phi}_{\mathrm{L}},\mathit{\Phi}_{\mathrm{R}})=(0,\mathit{\Phi}_{0}/2)$,
consistent with the earlier suppression of $\mathit{\Phi}_{\mathrm{L}}$
dependence as the gate voltages were lowered.

Switching currents and diode efficiency of Device 2 were also measured in the
switch-OFF configuration ($V_{\mathrm{J}}=-1.5~{}\mathrm{V}$), as shown in
Fig. S.8 for $V_{\mathrm{L}}=V_{\mathrm{R}}=-0.5~{}\mathrm{V}$. Oscillations
of both $I^{\pm}_{\mathrm{sw}}$ and $\eta$, suppressed along
$\mathit{\Phi}_{\mathrm{R}}$, were similar to those reported for Device 1 in
Figs. 2f–h of the Main Text. Interestingly, the maximum diode efficiencies of
approximately $12\%$ were significantly lower than in the corresponding
switch-ON case (Fig. S.7f), where $\eta^{\mathrm{max}}\approx 25\%$. Again,
this is attributed to the asymmetric supercurrent distribution in Device 2,
which led to small diode efficiencies along the
$\mathit{\Phi}_{\mathrm{L}}$-axis and required tuning of
$\mathit{\Phi}_{\mathrm{R}}$ to reach larger values in the case of Fig. S.7f .
When the switch was OFF, control over $\mathit{\Phi}_{\mathrm{R}}$ was
disabled, hence reducing $\eta^{\mathrm{max}}$.

The results shown for Device 2 were qualitatively captured by our extended
theoretical model, with parameters
$(\tau_{\mathrm{L}},\tau_{\mathrm{M}},\tau_{\mathrm{R}},T_{\mathrm{L}},T_{\mathrm{M}},T_{\mathrm{R}})$
set to $(0.91,0.87,0.93,3.5,2,6.5)$ for Figs. S.9a–c (Configuration 1), to
$(0.44,0.65,0.87,1,1,3.3)$ for Figs. S.9d–f (Configuration 2), and to
$(0,0.53,0.67,0.2,0.2,1)$ for Figs. S.9g–i (Configuration 3). Furthermore, we
simulated the device in the switch-OFF case using the same parameters as in
Configuration 2 and, in addition, $\tau_{\mathrm{RM}}=0.75$,
$T_{\mathrm{RM}}=4$; the result is displayed in Fig. S.10. In all cases, the
phase axes $(\phi_{\mathrm{L}},\phi_{\mathrm{R}})$ were converted to flux-
line-current axes $(I_{\mathrm{L}},I_{\mathrm{R}})$ using the transformation
described in Section 5. We note that the chosen parameters were such that
$\tau_{\mathrm{L}}<\tau_{\mathrm{R}}$ and $T_{\mathrm{L}}<T_{\mathrm{R}}$,
supporting our interpretation of the asymmetry between
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$ as a result of
asymmetric supercurrent flow in the device between terminals L and R. In
Configuration 3, the data was best described by using $\tau_{\mathrm{L}}=0$,
suggesting that the high-transmission channel between S and L was depleted.
The measurements performed on Device 2 and the simulations done within the
same model introduced for Device 1 support the generality of the observed
phenomena, and in particular corroborate the presence and origin of the JDE in
our device.

##  5 Current-to-Flux Remapping

The b-SQUID geometry of our devices enables control over two superconducting
phase differences (between terminals L and M, and between R and M), tuned by
the currents in the two flux-bias lines, $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$. These currents generated magnetic fluxes threading the two
superconducting loops, $\mathit{\Phi}_{\mathrm{L}}$ and
$\mathit{\Phi}_{\mathrm{R}}$, with a cross-coupling leading to an effect of
$I_{\mathrm{L(R)}}$ on the flux through the opposite loop,
$\mathit{\Phi}_{\mathrm{R(L)}}$. As a consequence, the
$\mathit{\Phi}_{\mathrm{L}}$\- and $\mathit{\Phi}_{\mathrm{R}}$-axes had a
finite slope with respect to the $I_{\mathrm{L}}$\- and $I_{\mathrm{R}}$-axes,
as visible in Fig. 2b of the Main Text. We describe the cross-coupling by
considering a mutual inductance matrix $\mathbf{M}$ that relates flux
$\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$ to flux-line
currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$:

$\begin{pmatrix}\mathit{\Phi}_{\mathrm{L}}\\\
\mathit{\Phi}_{\mathrm{R}}\end{pmatrix}=\mathbf{M}\cdot\begin{pmatrix}I_{\mathrm{L}}\\\
I_{\mathrm{R}}\end{pmatrix}=\begin{pmatrix}M_{\mathrm{LL}}&M_{\mathrm{LR}}\\\
M_{\mathrm{RL}}&M_{\mathrm{RR}}\end{pmatrix}\cdot\begin{pmatrix}I_{\mathrm{L}}\\\
I_{\mathrm{R}}\end{pmatrix}.$ (6)

To quantify $\mathbf{M}$ (for Device 1), we consider the
$\left(\mathit{\Phi}_{\mathrm{L}},\mathit{\Phi}_{\mathrm{R}}\right)$ and
$\left(I_{\mathrm{L}},I_{\mathrm{R}}\right)$ coordinates of two points of the
phase space [in addition to the origin,
$\left(\mathit{\Phi}_{\mathrm{L}},\mathit{\Phi}_{\mathrm{R}}\right)=\left(I_{\mathrm{L}},I_{\mathrm{R}}\right)=\left(0,0\right)$],
such as $\left(\mathit{\Phi}_{0},0\right)$ and
$\left(0,\mathit{\Phi}_{0}\right)$, and substitute them in Eq. 6 [47]. The
resulting $4\times 4$ equation system leads to the mutual inductance matrix:

$\mathbf{M}=\begin{pmatrix}6.98~{}\mathrm{pH}&-1.40~{}\mathrm{pH}\\\
-1.73~{}\mathrm{pH}&5.86~{}\mathrm{pH}\end{pmatrix}$ (7)

for Device 1, and

$\mathbf{M}=\begin{pmatrix}7.03~{}\mathrm{pH}&-1.39~{}\mathrm{pH}\\\
-1.67~{}\mathrm{pH}&5.80~{}\mathrm{pH}\end{pmatrix}$

for Device 2. We note that $\mathbf{M}$ is very similar between the two
devices, as the loop and flux-line geometry was lithographically identical. In
Fig. S.11, we apply $\mathbf{M}$ from Eq. 7 to perform a basis transformation
and plot the data presented in Figs. 2b–d (shown again in Figs. S.11a–c) as a
function of the magnetic fluxes $\mathit{\Phi}_{\mathrm{L}}$ and
$\mathit{\Phi}_{\mathrm{R}}$ (see Fig. S.11d–f).

The inverse transformation, obtained by inverting the mutual inductance
matrix, was employed to map the phase space
$(\phi_{\mathrm{L}},\phi_{\mathrm{R}})$ to the flux-line-current space
$(I_{\mathrm{L}},I_{\mathrm{R}})$ for the simulated data of Figs. 6 of the
Main Text, S.9 and S.10. In this transformation, the phase differences were
considered to be linearly related to the fluxes, thus
$\mathit{\Phi}_{\mathrm{L(R)}}=\mathit{\Phi}_{0}\times\phi_{\mathrm{L(R)}}/2\pi$
were substituted in Eq. 6 to obtain a direct relation between phases and
currents. This assumption is justified by noting that the inductance of the
superconducting loops, comprising geometric and kinetic contributions
$L_{\mathrm{loop}}=L_{\mathrm{geom}}+L_{\mathrm{k}}$, is negligible compared
to the Josephson inductance $L_{\mathrm{J}}$ existing between any pairs of
terminals in the 4TJJ. For each superconducting loop, the geometric inductance
is estimated as $L_{\mathrm{geom}}\approx 24~{}\mathrm{pH}$ [51], while the
kinetic inductance is calculated using the expression [52]:

$L_{\mathrm{k}}=\frac{l}{w}\frac{h}{2\pi^{2}}\frac{R_{\square}}{\mathit{\Delta}}\approx
100~{}\mathrm{pH},$ (8)

where $l=57~{}\mathrm{\mu m}$ is the length of the loop perimeter,
$w=1~{}\mathrm{\mu m}$ the width of the Al strip forming the loop, $h$ the
Planck’s constant, $R_{\square}\approx 1.5~{}\Omega$ the normal state
resistivity of the heterostructure stack (measured in Hall bar geometry where
the Al was not removed) and $\mathit{\Delta}\approx 180~{}\mathrm{\mu eV}$ is
the superconducting gap of Al. Geometric and kinetic contributions lead to a
total inductance $L_{\mathrm{loop}}\approx 124~{}\mathrm{pH}$. The Josephson
inductance between two superconducting terminals (for example, L and M when
considering the left loop) is estimated as
$L_{\mathrm{J}}=\mathit{\Phi}_{0}/2\pi I_{\mathrm{c}}\sim 3~{}\mathrm{nH}$,
where $I_{\mathrm{c}}\sim 100~{}\mathrm{nA}$ gives the order of magnitude of
the junction’s critical current.

Figure S.1: Phase-space linecuts at
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.1~{}\mathrm{V}$. (a) Switching current
$I^{+}_{\mathrm{sw}}$, measured for $I_{\mathrm{SD}}>0$, as a function of
flux-line currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, as in Fig. 2b of the
Main Text. Colored markers indicate the position of
$I_{\mathrm{R}}=100~{}\mathrm{\mu A}$ (green),
$I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ (white) and
$I_{\mathrm{R}}=-60~{}\mathrm{\mu A}$ (blue). (b–d) Differential resistance
$R$ as a function of $I_{\mathrm{L}}$ and $I_{\mathrm{SD}}$ for
$I_{\mathrm{R}}=100~{}\mathrm{\mu A}$ (b), $I_{\mathrm{R}}=20~{}\mathrm{\mu
A}$ (c) and $I_{\mathrm{R}}=-60~{}\mathrm{\mu A}$ (d), as indicated in (a).
Each map is obtained by merging two datasets recorded with $I_{\mathrm{SD}}$
ramping from $0$ to either positive or negative values [see white arrows in
(b)]. Figure S.2: Phase-space linecuts at
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.2~{}\mathrm{V}$. (a–c) Differential
resistance $R$ as a function of $I_{\mathrm{L}}$ and $I_{\mathrm{SD}}$, for
$V_{\mathrm{J}}=0$ (switch ON) and $I_{\mathrm{R}}=100~{}\mathrm{\mu A}$ (a),
$I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ (b) and $I_{\mathrm{R}}=-60~{}\mathrm{\mu
A}$ (c), as indicated in Fig. S.1a. In (a), a point where the switching
current reaches zero is indicated by the yellow arrow. (d) As in (b), but for
$V_{\mathrm{J}}=-1.5~{}\mathrm{V}$ (switch OFF). Figure S.3: Comparing
symmetric and asymmetric gate configurations. (a, b) Switching currents
$I^{+}_{\mathrm{sw}}$ and $I^{-}_{\mathrm{sw}}$ (respectively) as functions of
flux-line currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$ for symmetric gate
voltages $V_{\mathrm{L}}=V_{\mathrm{R}}=-0.3~{}\mathrm{V}$. The other gate
voltages were kept to $V_{\mathrm{S}}=0.1~{}\mathrm{V}$,
$V_{\mathrm{M}}=-0.15~{}\mathrm{V}$ and $V_{\mathrm{J}}=0$. The cyan arrows
indicate the switching current minima described in the text. (c)
Superconducting diode efficiency $\eta$ extracted from (a) and (b) as a
function of $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. Regions indicated by the
black arrows are discussed in the text. (d–f) As in (a–c), but for asymmetric
gate voltages $V_{\mathrm{L}}=-0.28~{}\mathrm{V}$ and
$V_{\mathrm{R}}=-0.31~{}\mathrm{V}$. In (d), yellow and blue markers indicate
the position of $I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ and
$I_{\mathrm{R}}=-140~{}\mathrm{\mu A}$, respectively. Figure S.4: Phase-space
linecuts at $V_{\mathrm{L}}=-0.28~{}\mathrm{V}$ and
$V_{\mathrm{R}}=-0.31~{}\mathrm{V}$. (a) Differential resistance $R$ as a
function of $I_{\mathrm{L}}$ and $I_{\mathrm{SD}}$ for $V_{\mathrm{J}}=0$ and
$I_{\mathrm{R}}=20~{}\mathrm{\mu A}$ (yellow marker in Fig. S.3d). (b) As in
(a), but for $I_{\mathrm{R}}=-140~{}\mathrm{\mu A}$ (blue marker in Fig.
S.3d). (c) As in (a), but for $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$ (switch OFF).
Each map is obtained by merging two datasets recorded with $I_{\mathrm{SD}}$
ramping from $0$ to either positive or negative values. Figure S.5: Results
for varying $V_{\mathrm{L}}$, $V_{\mathrm{R}}$ and extraction of diode
efficiencies. (a–e) Switching current $I^{+}_{\mathrm{sw}}$, measured for
$I_{\mathrm{SD}}>0$, as a function of flux-line currents $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$ for five settings of $V_{\mathrm{L}}=V_{\mathrm{R}}$,
indicated on the subfigures. (f–j) Diode efficiency extracted from (a–e) and
the corresponding measurements of $I^{-}_{\mathrm{sw}}$ (not shown),
respectively, as a function of $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. The
other gate voltages were kept to $V_{\mathrm{S}}=0.1~{}\mathrm{V}$,
$V_{\mathrm{M}}=-0.15~{}\mathrm{V}$ and $V_{\mathrm{J}}=0$. Figure S.6: Device
2: false-colored scanning electron micrograph of the four-terminal Josephson-
junction region. Color legend and labeled quantities are as in Figs. 1a and 1b
of the Main Text. Circuit layout and loop geometry of Device 2 are
lithographically identical to Device 1 (see Figs. 1a and 1b). Figure S.7:
Phase- and gate-tunable Josephson diode effect in Device 2. (a, b) Switching
currents $I^{+}_{\mathrm{sw}}$ and $I^{-}_{\mathrm{sw}}$, measured for
$I_{\mathrm{SD}}>0$ and $I_{\mathrm{SD}}<0$ respectively, as functions of
flux-line currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, for
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.4~{}\mathrm{V}$. Directions of the black
arrows in (a) indicate the periodicity axes, corresponding to the external
magnetic fluxes $\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$
threading the two superconducting loops, while their length (one period)
indicates the addition of one superconducting flux quantum $\mathit{\Phi}_{0}$
to the corresponding flux. (c) Superconducting diode efficiency $\eta$
obtained from (a) and (b) as a function of $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$. Features indicated by the orange and purple arrows are
discussed in the text. (d–f) As in (a–c), but for
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.5~{}\mathrm{V}$. (g–i) As in (a–c), but for
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.6~{}\mathrm{V}$. In all cases, the other
gate voltages were set to $V_{\mathrm{S}}=0.15~{}\mathrm{V}$,
$V_{\mathrm{M}}=-0.15~{}\mathrm{V}$ and $V_{\mathrm{J}}=0$. Figure S.8:
Josephson diode effect in Device 2 for $V_{\mathrm{J}}=-1.5~{}\mathrm{V}$
(switch OFF). (a, b) Switching currents $I^{+}_{\mathrm{sw}}$ and
$I^{-}_{\mathrm{sw}}$, measured for $I_{\mathrm{SD}}>0$ and
$I_{\mathrm{SD}}<0$ respectively, as functions of flux-line currents
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, for
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.4~{}\mathrm{V}$. Here,
$V_{\mathrm{L}}=V_{\mathrm{R}}=-0.5~{}\mathrm{V}$,
$V_{\mathrm{S}}=0.15~{}\mathrm{V}$ and $V_{\mathrm{M}}=-0.15~{}\mathrm{V}$.
(c) Superconducting diode efficiency $\eta$ obtained from (a) and (b) as a
function of $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$. Figure S.9: Simulations
with the extended model for Device 2. (a, b) Critical currents
$I^{+}_{\mathrm{c}}$ and $I^{-}_{\mathrm{c}}$, simulated for positive and
negative current bias (respectively), as functions of flux-line currents
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, using the parameters of Configuration 1
(see text). The model is described in the Main Text. Currents $I_{\mathrm{L}}$
and $I_{\mathrm{R}}$ were obtained from the superconducting phase differences
$\phi_{\mathrm{L}}$ (between terminals S and L) and $\phi_{\mathrm{R}}$
(between S and R) by applying a linear transformation (see Section 5 for
additional details). Phase axes $\phi_{\mathrm{L}}$ and $\phi_{\mathrm{R}}$,
that are the periodicity directions, are indicated by the black arrows, whose
length represents winding of the corresponding phase by $2\pi$. (c) Diode
efficiency $\eta$ extracted from (a) and (b) as a function of $I_{\mathrm{L}}$
and $I_{\mathrm{R}}$. (d–f) As in (a–c), but using the parameters of
Configuration 2. (g–i) As in (a–c), but using the parameters of Configuration
3. Figure S.10: Simulations with the extended model for Device 2 in the
switch-OFF case. (a, b) Critical currents $I^{+}_{\mathrm{c}}$ and
$I^{-}_{\mathrm{c}}$, simulated for positive and negative current bias
(respectively), as functions of flux-line currents $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$, using the parameters of Configuration 2 (see text) and
additional parameters $\tau_{\mathrm{RM}}=0.75$ and $T_{\mathrm{RM}}=4$. The
model is described in the Main Text. (c) Diode efficiency $\eta$ extracted
from (a) and (b) as a function of $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$.
Figure S.11: Linear transformation mapping the flux-line-current space into
the flux space. (a–c) Switching currents $I^{+}_{\mathrm{sw}}$ and
$I^{-}_{\mathrm{sw}}$ and diode efficiency $\eta$ as functions of flux-line
currents $I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, as in Figs. 2b–d of the Main
Text. (d–f) Same datasets of (a–c) plotted as functions of the external
magnetic fluxes $\mathit{\Phi}_{\mathrm{L}}$ and $\mathit{\Phi}_{\mathrm{R}}$,
upon applying the linear transformation described in the text.