# Diamagnetic mechanism of critical current non-reciprocity in multilayered
superconductors

Ananthesh Sundaresh  Department of Physics and Astronomy, Purdue University,
West Lafayette, IN 47907 USA  Jukka I. Väyrynen  Department of Physics and
Astronomy, Purdue University, West Lafayette, IN 47907 USA  Yuli Lyanda-Geller
Department of Physics and Astronomy, Purdue University, West Lafayette, IN
47907 USA  Leonid P. Rokhinson<EMAIL_ADDRESS>Department of Physics and
Astronomy, Purdue University, West Lafayette, IN 47907 USA  Department of
Electrical and Computer Engineering, Purdue University, West Lafayette, IN
47907 USA

Recent excitement in observation of non-reciprocal critical current (NRC) is
motivated by a suggestion that “superconducting diode effect” may be an
intrinsic property of non-centrosymmetric superconductors with strong spin-
orbit interactions[1]. Theoretically it has been understood that linear in the
Cooper pair momentum terms, caused by the Rashba spin-orbit and Zeemann
interactions or, more generally, any symmetry-allowed Lifshitz invariants[2]
in uniform singlet superconductors, do not contribute to the supercurrent,
although the role of higher-order terms remains unclear[3, 4]. In this work we
show that critical current non-reciprocity is a generic property of
multilayered superconductor structures in the presence of magnetic field-
generated diamagnetic currents. In the regime of an intermediate coupling
between the layers, the Josephson vortices are predicted to form at high
fields and currents. We report the observation of NRC in nanowires fabricated
from InAs/Al heterostructures. The effect is independent of the
crystallographic orientation of the wire, ruling out an intrinsic origin of
NRC. Non-monotonic NRC evolution with magnetic field is consistent with the
generation of diamagnetic currents and formation of the Josephson vortices.
This extrinsic NRC mechanism can be used to design novel devices for
superconducting circuits.

Diodes are the most basic elements of semiconductor electronics and
development of superconducting diodes can extend the functionality of the
superconducting circuitry. A non-reciprocal critical current (NRC) in a
multiply-connected superconductors is a well known effect and can be readily
observed in, e.g., an asymmetric superconducting rings [5]. An implicit
suggestion that NRC may be an intrinsic property of non-centrosymmetric
superconductors [1] generated a renewed theoretical and experimental interest
motivated by an analogy with the non-reciprocal resistivity due to the
magnetochiral effect, which can appear in uniform materials with broken
spatial and time-reversal symmetry [6]. However, a direct analogy between
corrections to resistivity and superconducting current is misleading because
the anisotropy of scattering is caused by the spin-orbit effects, while the
proposed origin of non-reciprocity in singlet-pairing superconductors is a
spin-independent Lifshitz invariant [7, 8]. It has been demonstrated in the
literature [2] that in uniform singlet superconductors in constant magnetic
field the Lifshitz invariants can be eliminated by a gauge (Galilean)
transformation from both the Ginzburg-Landau (GL) equation and the expression
for the supercurrent, so that linear in Cooper pair momentum terms do not lead
to non-reciprocity. Phenomenological treatment shows that cubic in the Cooper
pair momentum terms can lead to NRC corrections [3]. It has been also
suggested that the Rashba terms in the electron spectrum contribute to non-
reciprocity within the formalism of quasiclassical Eilenberger equations [9].
A symmetry analysis and microscopic calculations of the Cooperon propagator in
the presence of the Zeemann effect and a linear or cubic Dresselhaus spin-
orbit interactions in the electron spectrum show that the NRC magnitude and
sign depend on the crystallographic orientation of the supercurrent flow [10].
Such anisotropy should characterize non-reciprocity in
superconductor/semiconductor heterostructures and cubic uniform singlet
noncentrosymmetric superconductors. Another suggested mechanism of NRC is the
formation of non-uniform currents in superconducting multilayers [11, 12].

In this paper we show that NRC naturally arises in the presence of the
magnetic-field–generated diamagnetic currents when neighboring layers in
multilayer superconductors are strongly coupled. The total current through a
multilayer structure is divided between the layers as an inverse ratio of
their kinetic inductances. Initially only one of the layers reaches the
maximum (critical) current as the total current increases. A further current
increase in the superconducting state requires generating a phase difference
between the layers, which adds Josephson energy penalty to the total energy,
and in the case of strong interlayer coupling (large Josephson energy) the
whole system transitions into a normal state. Field-generated diamagnetic
currents will either increase or decrease an external current where transition
to the normal state occurs, thus leading to the NRC. In the regime of
intermediate interlayer coupling strengths, an interlayer phase difference can
change by $2\pi$, leading to the formation of Josephson vortices.
Experimentally, we report observation of NRC in nanowires fabricated from
InAs/Al heterostructures. The observed non-monotonic evolution of NRC with
magnetic field is consistent with the formation of Josephson vortices. Our
findings show that the extrinsic contribution to NRC is generic to multilayer
superconductors, and may provide a compelling explanation to the NRC observed
in Refs. 1 and 13, in the latter work magnetic flux produced by a Cr layer
generates opposite diamagnetic currents in the adjacent Nb and V layers.

The term “superconducting diode effect” has been used to describe NRC in
different systems, including thin superconducting films [14, 15, 16, 17, 18,
19] and Josephson junctions [20, 21, 22, 23, 24, 25, 26, 27, 28]. In the
former experiments the presence of out-of-plane magnetic field and formation
of vortices is essential for the observation of NRC, in this case the critical
current is determined by the strength and symmetry of the flux pinning
potential. In the latter case the critical current in Josephson junctions is
determined by the overlap of Andreev states. In this paper we restrict our
discussion to the origin of NRC in long nanowires, where critical current is
determined by the de-pairing velocity of Cooper pairs (the Bardeen limit
[29]).

Figure 1: Non-reciprocal critical current in Al/InAs nanowires. (a) Histograms
of switching currents for 10,000 positive $I_{sw}^{+}$ and negative
$I_{sw}^{-}$ current sweeps performed at $T=30$ mK and $B_{\bot}=100$ mT.
Inset shows a typical current-voltage characteristic. (b) Average switching
current for positive $\langle I_{sw}^{+}\rangle$ and negative $\langle
I_{sw}^{-}\rangle$ sweeps, non-reciprocal difference $\Delta I=\langle
I_{sw}^{+}\rangle-\langle I_{sw}^{-}\rangle$ and an average of all sweeps
$\langle I_{sw}\rangle$ is plotted as a function of in-plane magnetic field
$B_{\bot}$. In (c) enlarged $\Delta I$ data is colored to signify non-
monotonic field dependence and multiple sign changes. (d) Dependence of
$\Delta I$ on in-plane field orientation is measured at a constant $B=100$ mT.
Blue line is a fit with a sine function. Insert shows an AFM image of a $3\
\mu$m-long wire connected to wide contacts, yellow areas are Al, in darker
areas Al is removed and InAs is exposed.

We have studied switching currents $I_{sw}$ defining a transition from
superconducting to normal state in nanowires fabricated from
Al/InGaAs/InAs/InGaAs heterostructures [30], where patterned Al top layer
forms a nanowire and induces superconductivity in a high mobility InAs quantum
well via the proximity effect. An AFM micrograph of a typical device is shown
in the inset in Fig. 1. A typical current-voltage characteristic exhibits a
sharp switching transition limited by the current resolution ($<5$ nA for the
fastest sweep rates used in our experiments). A histogram of switching
currents $I_{sw}^{\pm}$ for positive (+) and negative (-) current sweeps is
shown in Fig. 1(a) for 10,000 sweeps. Field dependence of average values
$\langle I_{sw}^{+}\rangle$ and $\langle I_{sw}^{-}\rangle$ is plotted in Fig.
1(b) for the in-plane field $B_{\bot}$ perpendicular to the wire. The $\langle
I_{sw}^{+}\rangle$ and $\langle I_{sw}^{-}\rangle$ can be separated into a
symmetric $\langle I_{sw}\rangle=(\langle I_{sw}^{+}\rangle+\langle
I_{sw}^{-}\rangle)/2$ and asymmetric $\Delta I=\langle
I_{sw}^{+}\rangle-\langle I_{sw}^{-}\rangle$ parts, the latter being the non-
reciprocal component of the supercurrent. Both $\langle I_{sw}\rangle$ and
$\Delta I$ are non-monotonic functions of magnetic field. As shown in the
Supplement, a minima of $\langle I_{sw}\rangle$ at low fields vanishes above
350 mK (0.3 $T_{C}$), while there is no change in $\Delta I$ at least up to
750 mK ($>0.6T_{C}$). This difference in energy scales for the appearance of
NRC and non-monotonic evolution of $\langle I_{sw}\rangle$ indicates that
these are unrelated phenomena, and below we focus on the origin of NRC. Some
devices were fabricated with a top gate, which allows electrostatic control of
the electron density in the InAs layer not covered by Al; we found that
depletion of the 2D electron gas in the exposed InAs results in a slight
increase of $\langle I_{sw}\rangle$ but does not affect $\Delta I$. Similar
field effect has been observed previously in superconductor nanodevices [31]
and was attributed to the presence of quasiparticles [32], a conclusion
consistent with the observed gate dependence of the $\langle I_{sw}\rangle$.

Figure 2: Dependence of NRC on the nanowire length and crystallographic
orientation. (a) NRC is plotted for two 2 $\mu$m-long wires oriented along
$[110]$ and $[1\overline{1}0]$ crystallographic axes. Insets define mutual
orientataion of wires and fields. (b) NRC for 2, 3, and 5 $\mu$m wires. The
top and bottom curves are shifted vertically by 0.2 $\mu$A. Brackets with
arrows indicate a maximum $\Delta B$ needed to insert a flux $\phi_{0}=h/2e$
in the area defined by the corresponding wire lengths, as indicated by a
dashed loop in the inset. An effective length for the period marked by a
magenta bracket is $l=0.5\ \mu$m for the same loop.

Unlike the linear in Cooper pair momentum terms, higher order terms cannot be
removed by gauge transformation and it was shown that the presence of terms
$\sim\alpha_{3}Q^{3}\Delta^{2}$ cubic in the Cooper pair momentum in an
expansion of the Ginsburg-Landau coefficients can generally lead to non-zero
$\Delta I$ which is a non-monotonic function of $B$ and can even change sign
[3, 9] (here $\mathbf{Q}=-i\hbar\nabla-2e\mathbf{A}$ is a generalized Cooper
pair momentum, $\mathbf{A}$ is electromagnetic vector-potential). However, for
proximitized InAs layer, a generation of the terms higher order in the Cooper
pair momentum in the presence of the Rashba spin-orbit and Zeeman interactions
coexists with a similar generation of such terms due to the Dresselhaus spin-
orbit interactions. The importance of the Dresselhaus-like terms in the
electron spectrum is not limited to proximity structures, and they can play
significant role in any noncentrosymmetric material. Investigation of
realistic cubic terms in the Cooper pair momentum showed [10] that
nonreciprocity becomes highly anisotropic as a result of Dresselhaus-induced
contribution. For comparison with experiments, it is instructive to express
the odd in Cooper pair momentum part of the kinetic energy in coordinates
rotated by $\pi/2$ with respect to the principal crystallographic axes of
InAs, where $\hat{x}||[1\overline{1}0]$ and $\hat{y}||[110]$, see insert in
Fig. 2(a). In these coordinates, the cubic in the Cooper pair momentum kinetic
term originating from the cubic Dresselhaus electron spin-orbit interaction
reads

$f_{k}=|\kappa(B_{y}Q^{3}_{x}+B_{x}Q^{3}_{y}-Q_{x}Q_{y}[B_{x}Q_{y}+B_{y}Q_{x}])\Delta|^{2},$
(1)

where coefficient $\kappa$ contains the Dresselhaus constant $\beta_{D}$ and
other material parameters. The resulting NRC correction to the supercurrent is

$\Delta I\propto(B_{y}I_{x}^{2}+B_{x}I_{y}^{2}).$ (2)

This correction is independent of the sign of $I$ and is added or subtracted
to the $B=0$ current value depending on the direction of the current flow.
Here $B_{x}$ and $B_{y}$ enter symmetrically for wires oriented along $x$ and
$y$. However, in the configuration with the current $I\|\hat{x}$ and magnetic
field $B_{y}$ and the configuration with $I\|\hat{y}$ and $B_{x}$, this
expression has opposite signs for the same mutual orientation of $I$ and $B$,
see inset in Fig. 2(a). Thus, the Dresselhaus-induced contribution results in
NRC with opposite sign for wires oriented along $[1\overline{1}0]$ and $[110]$
crystallographic axis. The cubic (and generally all odd) in Cooper pair
momentum terms originating from the Rashba electronic interactions, when added
with the Dresselhaus-induced terms, will produce anisotropy in the absolute
value of NRC, and, in particular, different values of non-reciprocal
asymmetrical component of the current for those two directions. Theoretical
investigation of electronic spectra of these systems [33] suggests that in
narrow InAs quantum wells cubic Dresselhaus terms are larger than the Rashba
terms. The lower limit for the value of the Dresselhaus contribution can be
extracted from the total spin-orbit anisotropy (which is defined by the ratio
between a linear Rashba, and a linear and cubic Dresselhaus terms in
electronic spectrum), which was measured to be 70% in spin-galvanic and
circular photogalvanic experiments [34] and $>15\%$ in transport
experiments[35, 36]. Such anisotropies must result in the corresponding
crystallographic anisotropy of the NRC, which is not observed in our
experiments, Fig. 2(a). Therefore, we conclude that the NRC we observed is not
intrinsic. The observed NRC does not depend on the wire length, Fig. 2b, which
rules out trivial effects related to the formation of spurious loops due to
the presence of wire/contact boundaries.

Figure 3: NRC in an asymmetric superconducting loop. (a) An average switching
current for positive $\langle I_{sw}^{+}\rangle$ and negative $\langle
I_{sw}^{-}\rangle$ sweeps, non-reciprocal difference $\Delta I=\langle
I_{sw}^{+}\rangle+\langle I_{sw}^{-}\rangle$ and an average of all sweeps
$\langle I_{sw}\rangle$ plotted as a function of out-of-plane magnetic field
$B_{o}$ for a loop shown in the insert in (b). Note that $\langle
I_{sw}\rangle$ is maximal while $\Delta I=0$ when the flux $\phi=n\phi_{0}$.
In (b) $\Delta I$ for the nanowire and the loop are plotted together as a
function of a reduced flux $\phi/\phi_{0}$, where we used $S_{wire}=0.0052\
\mu$m2 for the effective area in the wire and $S_{loop}=2.59\ \mu$m2 in the
loop.

While recent interest in NRC has been motivated by a possibility of the
intrinsic origin of the effect, NRC naturally arises in multiply-connected
superconductors. In superconducting loops, the critical current is modulated
by an external flux $\phi=BS_{loop}$ piercing the loop. In a loop with
asymmetric arms, the current maximum is shifted from $B=0$, and the sign of
the shift depends on the direction of the current as shown in Fig. 3(a). A
non-reciprocal component of the switching current $\Delta I$ is linear in $B$
in the vicinity of $B=0$, reaches extrema at $\phi\approx\phi_{0}/4$, changes
sign and oscillates with a period $\Delta\phi=\phi_{0}$. Thus, an asymmetric
loop is the simplest “superconducting diode”. There is a clear similarity
between $\Delta I$ measured in an asymmetric superconducting loop and in an
Al/InAs nanowire as emphasized in Fig. 3(b), suggesting that non-monotonic NRC
in our nanowires may be due to emerging current loops.

Figure 4: Non-reciprocity of the critical current in the presence of
diamagnetic currents. Diamagnetic currents in (a) a uniform superconductor and
(b) a heterostructure. (c) A two-layer heterostructure is modeled as two zero
thickness wires separated by a distance $d$ with coupling described by the
Josephson energy $\mathcal{E}_{J}$. (d) Schematic of current distribution
between the wires $I_{i}=I_{0i}-(-1)^{i}I_{dia}$, $i=1,2$, and the phase
difference $\Delta\phi$ as a function of an external current
$I_{ext}=I_{1}+I_{2}$ for $B=0$ (solid lines), $B>0$ (dotted lines) and $B<0$
(dashed lines). For weakly coupled wires $\mathcal{E}_{J}\ll\mathcal{E}_{k}$,
the critical current is field-independent $I^{{}^{\prime}}_{c}=I_{c1}+I_{c2}$,
see the text; the critical current is reduced and acquires a linear-in-$B$
correction in a strong coupling regime $\mathcal{E}_{J}\gg\mathcal{E}_{k}$ due
to the phase locking $\Delta\phi=0$. (e) In the intermediate coupling regime
$\mathcal{E}_{J}\sim\mathcal{E}_{k}$ Josephson vortices may form generating a
$2\pi$ phase twist, in this case NRC becomes a non-monotonic function of $B$.
(f) Calculated NRC $\Delta I$ is plotted as a function of flux
$\Phi=S_{v}B_{y}$ for several $B_{x}$, Eq. (S12).

External magnetic field generates circular diamagnetic currents in a
superconductor, as shown schematically in Fig. 4(a,b), and these currents
affect the critical current. In homogeneous superconductors the presence of
diamagnetic currents will not result in the critical current non-reciprocity,
but in a heterogeneous superconductor, in general, their presence will lead to
NRC. Qualitatively, the origin of NRC can be understood from a simplified
model of a superconductor heterostructure represented as two coupled zero
thickness superconducting wires separated by a distance $d$, Fig. 4(c). The
total energy of the two-wire system can be written as a sum of kinetic and
Josephson energies,

$E_{\text{tot}}=\int
dx\left[\mathcal{E}_{k}-\mathcal{E}_{J}\cos(\Delta\phi)\right]\,,$ (3)

where $\mathcal{E}_{k}=L_{1}I_{1}^{2}+L_{2}I_{2}^{2}$, $\mathcal{E}_{J}$ is
the Josephson coupling, $\Delta\phi=\phi_{1}(x)-\phi_{2}(x)$ is the phase
difference between superconducting condensates, and $L_{i}$ are the kinetic
inductances per unit length in wires labeled by an index $i=1,2$.
Supercurrents in each wire
$I_{i}=(2eL_{i})^{-1}(\hbar\partial_{x}\phi_{i}-2eA_{x})$ should satisfy
charge conservation constraint $I_{1}(x)+I_{2}(x)=I_{ext}$, where $I_{ext}$ is
the applied external current. Detailed solution for this model can be found in
the Supplementary Materials, and we outline now the main results. For small
external currents ($I_{1}<I_{c1}$ and $I_{2}<I_{c2}$, where $I_{ci}$ are the
critical currents in the wires) it is energetically favorable to keep the
phase difference $\Delta\phi$ constant ($\Delta\phi=0$ for
$\mathcal{E}_{J}>0$). Then, the currents can be expressed as
$I_{1}=I_{01}+I_{dia}$ and $I_{2}=I_{02}-I_{dia}$, where $I_{01},I_{02}\propto
I_{ext}$ with $I_{01}/I_{02}=L_{2}/L_{1}=\eta^{-1}$ and
$I_{dia}=B_{y}d/(L_{1}+L_{2})$. Dependence of $I_{1}$ and $I_{2}$ on $I_{ext}$
for $B_{y}>0$, $B_{y}=0$ and $B_{y}<0$ is plotted schematically in Fig. 4(d).
As $I_{ext}$ increases and one of the currents ($I_{1}$ in our example)
reaches the critical value $I_{c1}$, further external current increase
requires an increase of $|\Delta\phi|$ because the excess current has to flow
through the remaining superconducting wire with the current $I_{2}$. In the
case of weak interwire coupling, $\mathcal{E}_{J}\ll\mathcal{E}_{k}$,
deviation of $\Delta\phi$ from zero does not lead to a significant energy
penalty and the critical current of the whole system
$I^{\prime}_{c}=I_{c1}+I_{c2}$ does not depend on the magnetic field
direction. In the opposite regime of strong coupling,
$\mathcal{E}_{J}\gg\mathcal{E}_{k}$, the energy cost associated with the
formation of Josephson currents (the last term in Eq. (3)) is prohibitively
high and the whole system transitions to a normal state at
$I_{ext}\approx(1+\eta)(I_{c1}-I_{dia})$, resulting in $\Delta
I=-2(\eta+1)I_{dia}(B_{y})$ (this equation is correct for
$\beta>\eta+(\eta+1)I_{dia}/I_{c1}$, where $\beta=I_{c2}/I_{c1}$, NRC for
other scenarios is listed in the Supplementary Materials). Thus a
superconducting diode effect is a generic property of coupled multilayer
superconductors.

As $B_{y}$ and diamagnetic currents increase, the phase locking condition
$\Delta\phi=0$ along the length of the wires leads to a significant increase
of $\mathcal{E}_{k}$. At a critical field $B_{c}=(3/\pi^{2})\Phi_{0}/(l_{J}d)$
it becomes energetically favorable to reduce the overall energy by twisting
the phase difference by $2\pi$ locally forming a Josephson vortex
($l_{J}\approx\Phi_{0}/(2\pi\sqrt{2\mathcal{E}_{J}L_{2}})$ and $\Phi_{0}=h/2e$
is the flux quantum). Evolution of the phase difference
$\Delta\phi(x)=4\arctan[\exp(x/l_{J})]$, vortex-induced currents in the wires
$I_{v}(x)$, and interwire Josephson current density $j_{J}(x)$ across a vortex
are shown schematically in Fig. 4(e). The maximum of $I_{v}(x)$ at the center
of the vortex determines the Josephson vortex contribution to NRC. In the
absence of quantum fluctuations formation of a vortex is accompanied by an
abrupt re-distribution of currents between the wires, which results in a
sawtooth NRC dependence on magnetic field. Generation of multiple Josephson
vortices does not modify NRC compared to a single vortex case unless the
vortices significantly overlap so that the maximum of $I_{v}(x)$ exceeds its
single-vortex value.

In Fig. 4(f) we plot $\Delta I(B_{y})$ for several $B_{x}$ using Eq. (S12) in
the Supplemental Material. A gradual change of $\Delta I$ near $\Phi_{0}/2$ is
due to quantum fluctuations of the winding number due to strong coupling of
the vortex to current-carrying wires. This smearing is similar to the gradual
change of a critical current in a ring connected to superconducting leads
(Fig. 3), as compared to an abrupt reversal of persistent currents at
$\Phi_{0}/2$ in isolated rings [37]. The period of oscillations of $\Delta I$
corresponds to the flux threading an effective vortex area
$S_{v}=(\pi^{2}/3)l_{J}d=l_{v}d$. The period $\Delta B_{\bot}=400$ mT
translates into the length $l_{v}\approx 500\mathrm{nm}$, where $\Delta\phi$
substantially deviates from zero. We estimate
$l_{v}<\xi_{InAs}=\sqrt{\xi^{0}_{InAs}l^{m}_{InAs}}\approx 750$ nm and expect
proximity-induced superconductivity in InAs to be preserved in the presence of
a vortex. Here we use $\xi_{InAs}^{0}=\hbar v_{F}/\pi\Delta^{*}\approx 1.8\
\mu$m, induced gap in InAs $\Delta^{*}\approx\Delta=1.796k_{B}T_{c}=230\mu eV$
(induced gap is close to the gap of Al in these heterostructures [38]), and
the mean free path in uncapped InAs 2D gas $l^{m}_{InAs}\approx 300$ nm.

Finally, we use the two-wire model to estimate the temperature and in-plane
field $B_{\|}\|\hat{x}$ dependences of NRC assuming that both parameters
affect the Cooper pair density $n_{2}$ in InAs. In the vicinity of
$B_{\bot}=0$ the amplitude of $\Delta I\propto L_{2}^{-1}\propto n_{2}$ and is
expected to decrease with an increase of $T$ or $B_{\|}$. The critical field
$B_{c}\propto\sqrt{\mathcal{E}_{J}L_{2}}$ depends on $\mathcal{E}_{J}\propto
n_{2}$, and the period of oscillations is expected to be $T$\- and
$B_{\|}$-independent, Fig. 4(d). Josephson coupling $\mathcal{E}_{J}$ is
exponentially sensitive to the thickness of the InGaAs spacer and we expect
slight variations of the period $\Delta B_{\bot}$ between the samples. These
qualitative estimates are consistent with experimental observations, see Figs.
S2 and S3 in the Supplemental Material.

## Methods

Materials. The wafer was grown using Molecular Beam Epitaxy (MBE) on an InP
substrate. The heterostructure consists of 1 $\mu$m graded InxAl1-xAs
insulating buffer followed by a
In0.75Ga0.25As(4nm)/InAs(7nm)/In0.75Ga0.25As(10nm) multilayer structure capped
in-situ with 7nm of Al. The two-dimensional electron gas has a peak mobility
of 28000 cm2/Vs at a density $8\times 10^{11}$ cm-2.

Sample Fabrication. The nanowires were fabricated using standard electron beam
lithography. The mesas were defined by first removing the top Al layer with Al
etchant Transene D and then a deep wet etching using H3PO4:H2O2:H2O:C6H8O7
(1ml:8ml:85ml:2g). Nanowires are defined in the second step of lithography by
patterning the Al layer. Some devices have a top electrostatic gate, in these
devices a 20 nm HfO2 is grown by atomic layer deposition followed by a
deposition of a Ti/Au (10/100 nm) gate.

Measurements. Current-voltage sweeps were performed using a homemade high
speed high resolution DAC/ADC (digital-to-analog and analog-to-digital
converter) setup. The sweeps were automatically interrupted at the
superconductor-normal transition ($I_{sw}$) in order to minimize device
heating. Current sweep rate and delay between sweeps have been optimized to
obtain $<5$ nA current resolution and to keep device temperature $<50$ mK at
the base temperature of the fridge. The data has been corrected for an
instrumental cooldown-dependent constant current offset (generated in the
circuit by uncompensated voltages in the system and limited by a 100 k$\Omega$
current source resistor) to insure that $\Delta I=0$ at $B=0$.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgments

The Al/InAs heterostructures were provided by Michael Manfra Group at Purdue
University. The authors thank Lev Ioffe and Igor Aleiner for stimulating
discussions. Experimental part was initially supported by the U.S. Department
of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering under Award DE-SC0008630, the work was completed with the support
by NSF award DMR-DMR-2005092 (A.S. and L.P.R.). Theoretical work is supported
by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering under Award DE-SC0010544 (Y.L-G) and the
Office of the Under Secretary of Defense for Research and Engineering under
award number FA9550-22-1-0354 (J.I.V.).

## Author contributions

L.P.R conceived, A.S. performed experiments, J.I.V. and Y.L.G developed the
theory. All authors participated in writing the manuscript.

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Diamagnetic mechanism of critical current non-reciprocity in heterostructured
superconductors
Supplementary Materials

Ananthesh Sundaresh, Jukka I. Väyrynen, Yuli Lyanda-Geller and Leonid P.
Rokhinson

###### Contents

  1. I Comparison of symmetric and asymmetric contributions to the critical current.
  2. II Dependence of NRC on Temperature.
  3. III Dependence of NRC on $B_{\|}$.
  4. IV NRC in nanowires of various width
  5. V Gate dependence of NRC and critical current.
  6. VI Absence of NRC in aluminum nanowire.
  7. VII In-plane magnetic field alignment
  8. VIII Theory: Geometric effects and $I_{c}$ non-reciprocity in coupled superconducting wires
  9. IX Determining total critical current in a 2 wire model

##  I Comparison of symmetric and asymmetric contributions to the critical
current.

Figure S1: Field dependence of symmetric and asymmetric parts of the switching
current. (a,b) Temperature dependence of $\Delta I$ and $\langle
I_{sw}\rangle$ measured as a function of $B_{\bot}$ ($\theta=90$ deg). The
curves are offset by -0.1 $\mu$A ($\Delta I$) and 0.01 ($\langle
I_{sw}\rangle$). (c,d) Angle dependence is measured by rotating magnetic field
of constant magnitude $B=100$ mT at the base temperature.

The switching current can be decomposed into symmetric $\langle
I_{sw}\rangle=(\langle I_{sw}^{+}\rangle+\langle I_{sw}^{-}\rangle)/2$ and
asymmetric $\Delta I=\langle I_{sw}^{+}\rangle-\langle I_{sw}^{-}\rangle$
parts. Their dependence on magnetic field is plotted in Fig. S1. Both $\langle
I_{sw}\rangle$ and $\Delta I$ are non-monotonic functions of $B_{\bot}$,
however they have very different $T$\- and field-angle-dependencies. $\Delta
I$ is almost unaffected by temperature up to $T\sim 0.6T_{c}$, while a dip
around $B_{\bot}=0$ in $\langle I_{sw}\rangle$ is developed at $T<0.3T_{c}$.
At constant $B=100$ mT, the angular dependences are $\Delta
I\propto\sin(\theta)$, but field-dependent correction to $\langle
I_{sw}\rangle$ is $\propto\cos(2\theta)$. These differences in energy scales
($T$-dependence) and angular dependencies indicate that suppression of
$\langle I_{sw}\rangle$ near $B=0$ and asymmetric $\Delta I$ have different
physical origins. Indeed, suppression of a critical current near $B=0$ has
been reported in previous works on single-layer nanowires and was attributed
to the presence of quasiparticles and/or magnetic impurities [32, 39], which
differ from geometrical effects responsible for $\Delta I(B)$ dependence.

##  II Dependence of NRC on Temperature.

Figure S2: The effect of temperature on NRC. (a) The amplitude of NRC $[\Delta
I(-100mT)-\Delta I(100mT)]$, (b) the standard deviation of the switching
currents at $B_{\perp}$ = 0, and (c) the average switching current at
$B_{\perp}$ = 0 are plotted as a function of the reduced temperature. NRC
amplitude follows the T-dependence of the Cooper pair density $n_{s}(T)$,
consistent with Eq. (S13). $\langle I_{sw}\rangle(T)$ follows the Bardeen
relation[29].

##  III Dependence of NRC on $B_{\|}$.

Non-reciprocity of the switching current is linearly suppressed by an in-plane
magnetic field $B_{\|}\|I$ and vanishes at $\approx 750$ mT. Within the same
range of $B_{\|}$ the magnitude of the switching current remains almost
constant (decreases $<2.5\%$ at $B_{\|}=750$ mT).

Figure S3: The effect of an in-plane current $B||I$ on the non-reciprocal
supercurrent. (a) Evolution of $\Delta I$ in the presence of $B_{\|}$. The
plots are vertically shifted for clarity (b) The NRC amplitude falls
approximately linearly with $|B_{\|}|$ (c) Dependence of average switching
current $\langle I_{sw}\rangle$ at $B_{\bot}=0$ on $B_{\|}$. All data is taken
at the base temperature.

##  IV NRC in nanowires of various width

We have studied NRC in several nanowires of different width and length. Since
all devices were fabricated from similar wafers, the Josephson coupling and,
therefore, $l_{J}$ are similar in all devices, and we expect the amplitude of
$\Delta I$ and period $\Delta B$ to be similar. Indeed, that is the case for
most devices, see Fig. S4a. One nanowire showed $\approx 2\times$ enhancement
of $\Delta I$ and $\approx 2\times$ reduction of $\Delta B$, which would be
consistent with a local enhancement of $l_{J}$ by a factor of 2.

Figure S4: NRC in other devices. (a) NRC in nanowires of different width. The
plots are shifted vertically for clarity. (b) Multiple sign reversal of
$\Delta I$ in another nanowire. (c) one out of $\sim 20$ nanowires fabricated
from similar wafers showed enhanced magnitude of $\Delta I$ and reduced
$\Delta B$.

##  V Gate dependence of NRC and critical current.

On one of the samples we fabricated an electrostatic gate which covered the
wire and a surrounding InAs 2D gas. In order to deplete electrons in InAs in
the regions where it is not screened by Al, we apply a large negative gate
voltage. We see no observable effect on the NRC when varying the gate voltage.
InAs is expected to be fully depleted for applied gate voltage -1.5V. We
measured NRC at different gate voltages varying from 0 to -4.5V and observed
no variation of $\alpha=d\Delta I/dB_{\bot}$ near $B=0$ or $\Delta B$. Slight
(up to 0.26%) increase of $\langle I_{sw}\rangle$ at large negative gate
voltages is observed. Negative gate voltage also depletes carriers in Al
(albeit their negligibly small fraction) and, thus, should result in the
decrease of $I_{c}$, contrary to the observed increase. The observed increase
of the switching current may result from the reduction of quantum fluctuations
due to the reduction of InAs volume for Cooper pairs to enter and, as a
consequence, increasing switching current to be closer to the value of the
critical current.

Figure S5: The effect of gate voltage on NRC. (a) NRC shows no observable
difference on varying the gate voltage. (b) $\alpha$ shows very little
variation with gate voltage. (c) When a negative gate voltage is applied the
$\langle I_{sw}\rangle$ increases.

##  VI Absence of NRC in aluminum nanowire.

Figure S6: No NRC in a control device. A control 150 nm wide and 3 $\mu$m long
nanowire is fabricated from a 20 nm thick Al film deposited on a semi-
insulating Si wafer. This device shows no NRC.

##  VII In-plane magnetic field alignment

Figure S7: (a) Field dependence of $\langle I_{sw}\rangle$ shows a Meissner
state up to $B_{z}\approx 18$ mT. (b) No NRC is observed in an out-of- plane
magnetic field.

Magnetic fields are generated by a 3-axis vector magnet. The critical out-of-
plane field for our wires is $B^{z}_{c2}\approx 60$ mT. A sharp reduction of
$\langle I_{sw}\rangle$ at $B_{z}>18$ mT is associated with an entrance of
Abrikosov vortices. In order to align the in-plane field with the plane of the
sample the following alignment procedure has been used. The in-plane field was
ramped to $B_{\|}^{{}^{\prime}}\approx 800$ mT, beyond the field where NRC is
observed. Next, $B_{z}$ field is scanned $\pm 30$ mT and a symmetry point
$B_{z}^{{}^{\prime}}$ is determined. In subsequent scans a linear correction
$B_{z}=aB_{\|}$, where $a=B_{z}^{{}^{\prime}}/B_{\|}^{{}^{\prime}}$, is
applied to keep $B_{\|}$ aligned with the sample plane with a precision of
$<0.1$ degree.

##  VIII Theory: Geometric effects and $I_{c}$ non-reciprocity in coupled
superconducting wires

Figure S8:  (a) Schematic picture of the model to explain non-reciprocity. The
dark grey regions depict two superconducting wires labeled 1 and 2
(corresponding to Al and proximitized InAs wires, respectively) with the order
parameter phases $\phi_{1}$, $\phi_{2}$. The region between the wires denotes
the insulating barrier of thickness $d$. In most positions $x$, the phases are
locked to $\phi_{1}=\phi_{2}\,(\mathrm{mod}\,2\pi)$ due to a strong Josephson
coupling. In the region of length $l_{v}$ spanned by the Josephson vortex the
phases are not equal and as a result the phase difference winds by an
additional $2\pi n$ over the vortex. The vertical arrows denote the resulting
Josephson currents flowing between the two wires in the vortex. (b) Total
energy vs magnetic field in the two-wire model. The dashed curves show the
spectrum obtained from Eq. (S7). The three parabolas correspond to Josephson
vortices/antivortices with $n=-1,0,1$. The solid curves show the energies when
coherent vortex tunneling (strength $E_{t}=0.4$ in units of
$\frac{\eta}{1+\eta}\frac{\hbar^{2}}{e^{2}L_{1}l_{J}}$) is included, leading
to avoided crossings of states with different $n$.

In this section we derive the critical superconducting current in two
Josephson-coupled superconducting wires (the “two-wire model”) in the presence
of an external magnetic field. We show that at high enough magnetic field, it
becomes energetically favorable to form a Josephson vortex (Fig. S8a), which
in turn can lead to an oscillatory non-reciprocity of the critical current
(Fig. 4 of the main text). Furthermore, the oscillations will be damped due to
one of the wires turning normal upon increasing the magnetic field.

Let us consider a pair of parallel superconducting wires 1 (Al wire) and 2
(proximitized InAs) along the $x$-direction with a magnetic field $B_{\perp}$
in the $y$-direction, normal to the plane containing two wires. The
corresponding vector potential is $A_{x}(z)=B_{\perp}z$ with the two wires
separated by distance $d$ being at positions $z=z_{1,2}=-(-1)^{1,2}d/2$, see
Fig. S8a. We ignore here the screening of the magnetic field by the Josephson
vortex; this effect would merely modify the Josephson length $l_{J}$
(introduced below). We also approximate the wires as one-dimensional, given
that their typical thickness is smaller than the penetration depth and the
width is smaller that the size of the Peal vortex. This makes the supercurrent
distribution approximately uniform within the wire. Denoting $\phi_{1,2}$ the
phases of the superconducting order parameters, we have a supercurrent in wire
$i$ given by

$I_{i}=\frac{1}{2eL_{i}}\left(\hbar\partial_{x}\phi_{i}-2eA_{x}(x,z_{i})\right)\,,$
(S1)

in terms of the kinetic inductances (per length)
$L_{i}=m_{i}/(e^{2}S_{i}n_{i})$ for wires $i=1,2$. Here $S_{i}$, $m_{i}$, and
$n_{i}$ denote the cross-sectional area, the effective mass, and the Cooper
pair densities. For Al wire ($i=1$) we will account for disorder by
multiplying $n_{i}$ by $\sqrt{l/\xi}$, where $l\approx 2\mathrm{nm}$ is the
mean free path and $\xi\approx 1\mu\mathrm{m}$ is the coherence length [29].
Thus, we use $L_{1}\to L_{1}\sqrt{\xi/l}$ in our final estimates.

The phases $\phi_{1,2}(x)$ can be found by minimizing the total energy

$E_{\text{tot}}=\int
dx\left[\frac{1}{2}L_{1}I_{1}^{2}+\frac{1}{2}L_{2}I_{2}^{2}-\mathcal{E}_{J}\cos(\phi_{1}-\phi_{2})\right]\,,$
(S2)

that includes kinetic energies of each wire and a Josephson energy density
$\mathcal{E}_{J}$ coupling the two wires. In the presence of an applied
external supercurrent $I_{ext}$, there is a constraint
$I_{1}(x)+I_{2}(x)=I_{ext}$ at every point $x$. The constrained energy
minimization leads to the Sine-Gordon equation for
$\varphi=\phi_{1}-\phi_{2}$,

$\frac{\partial^{2}\varphi}{\partial x^{2}}=l_{J}^{-2}\sin\varphi\,,$ (S3)

where $l_{J}=1/\sqrt{8e^{2}\mathcal{E}_{J}(L_{1}+L_{2})/\hbar^{2}}$ is the
Josephson length that determines the characteristic size of a Josephson
vortex. We now solve Eq. (S3) with the appropriate boundary conditions. We
assume that the Josephson coupling in Eq. (S2) is strong, such that
$\phi_{1}=\phi_{2}\,(\mathrm{mod}\,2\pi)$ for most $x$. If the two phases were
locked for _all_ $x$, i.e. $\varphi(x)=0\,(\mathrm{mod}\,2\pi)$, we would find
a non-reciprocal critical current $I_{c}(B_{\perp})$ with the non-reciprocity
$\Delta I=I_{c,+}-I_{c,-}$ that increases monotonically with $B_{\perp}$.
Experimentally, a non-monotonic dependence is observed, see Fig. 1b.

The non-monotonic $\Delta I$ can be explained by a formation of a Josephson
vortex, see Fig. 4. In the Josephson vortex, the phase difference $\varphi$
increases by $2\pi$ approximately over the distance $2\pi l_{J}$; explicitly,
$\varphi(x)=4\arctan e^{x/l_{J}}$ for a vortex at $x=0$.

The Josephson vortex solution yields a current distribution

$\displaystyle I_{1}(x)$ $\displaystyle=\frac{1}{1+\eta}I_{ext}+\delta
I_{n}(x)\,,$ (S4) $\displaystyle I_{2}(x)$
$\displaystyle=\frac{\eta}{1+\eta}I_{ext}-\delta I_{n}(x)\,,$ (S5)
$\displaystyle\delta I_{n}(x)$
$\displaystyle=\frac{2\eta}{1+\eta}\frac{1}{2eL_{1}}\frac{\hbar}{l_{J}}\left(n\,\text{sech}\frac{x}{l_{J}}-\frac{3}{\pi}\frac{\Phi}{\Phi_{0}}\right)\,,$
(S6)

where we introduced an integer index $n$, $n=\pm 1$ for the Josephson
vortex/antivortex and $n=0$ in the absence of the vortex. The vortex is
centered at $x=0$, which also turns out to be the position of the maximal
circulating currents in the wires 1 and 2. We denote
$\eta=L_{1}/L_{2}=S_{2}\frac{n_{2}}{m_{2}}/S_{1}\frac{n_{1}}{m_{1}}$ and
introduce the flux $\Phi/\Phi_{0}=S_{v}B_{\perp}/(\pi\hbar/e)$ through the
effective vortex area $S_{v}=(\pi^{2}/3)l_{J}d$.

The formation of the Josephson vortex becomes energetically favorable at a
large enough magnetic field $B_{\perp}$. The energy cost is determined from
Eq. (S2) by the balance of the Josephson energy $E_{J}$ lost and the kinetic
energy gained in the creation of a vortex. Ignoring $n$-independent terms, we
find (see Fig. S8b),

$E_{\text{Vortex}}(n)=\frac{\eta}{1+\eta}\frac{\hbar^{2}}{e^{2}L_{1}l_{J}}\left[\left(n-\frac{3}{2}\frac{\Phi}{\Phi_{0}}\right)^{2}+\frac{1}{2}|n|\right]$
(S7)

where $n=0,\pm 1$. This energy is analogous to the (inductive) energy of a
superconducting ring with a phase winding $2\pi n$ [40] apart from the last
term in Eq. (S7) which is the cost in Josephson energy. In the absence of
quantum fluctuations and at $T=0$, one finds from Eq. (S7) that the thermal
average $\langle n\rangle=[\Phi/\Phi_{0}]$ is given by the nearest integer to
$\Phi/\Phi_{0}$, leading to a sawtooth-like dependence for $\Delta I$ versus
$B_{\perp}$ (see below). Fluctuations will smear out the sawtooth dependence.
In analogy to a superconducting ring [40], we expect to find a harmonic
dependence on the flux on a linear background in the case of strong quantum or
thermal fluctuations,

$\langle n\rangle=\frac{\Phi}{\Phi_{0}}-\delta
n\sin\frac{2\pi\Phi}{\Phi_{0}}\,,$ (S8)

where $\delta n\ll 1$ due to strong fluctuations. Importantly, in the case of
quantum fluctuations, $\delta n$ is independent of the temperature, whereas
for thermal fluctuations one has exponential dependence on $1/T$. As we
discuss below, the harmonic dependence on the flux translates to a similar
dependence in the non-reciprocal part $\Delta I$ of the critical current, in
agreement with experimental data. The observed weak $T$-dependence in Fig. S1a
indicates that quantum fluctuations exceed thermal fluctuations in the
experiment.

Figure S9:  Left:  The critical current non-reciprocity $\Delta I$, Eq. (S12),
versus the flux $\Phi=B_{\perp}dl_{v}$ through the Josephson vortex. The
crosses correspond to the approximation, Eq. (S13). The applied field
$B_{\parallel}$ suppresses the proximity effect and therefore $\Delta I$. In
the figure $E_{t}=0.4$ in units of
$\frac{\eta}{1+\eta}\frac{\hbar^{2}}{e^{2}L_{1}l_{J}}$. Right: $\Delta I$ (at
$B_{\parallel}=0$) for different strengths $E_{t}$ (in units of
$\frac{\eta}{1+\eta}\frac{\hbar^{2}}{e^{2}L_{1}l_{J}}$) of coherent vortex
tunneling that controls vortex number fluctuations; weak tunneling leads to a
sawtooth-like $\Delta I$.

The critical current through our two wire system with contacts, effectively
forming a ring-like structure is determined by the condition that at large
enough $I_{ext}$, one of the wires ( arms of the ring) turns normal.
(Experiment indicates that the switching happens in Al, i.e., wire 1, see
below.) Assuming that $I_{ext}>0$ and the wire 1 turns normal, the
corresponding condition is $I_{ext}=I_{c,+}$, where

$I_{c,\pm}=(1+\eta)(\pm I_{1,c}-\delta I)\,,$ (S9)

$I_{1,c}$ is the critical current of wire 1 and $\delta I=\langle\delta
I_{n}(0)\rangle$ is the circulating current at its peak value at $x=0$.
Likewise, for $I_{ext}<0$ we find $I_{ext}=I_{c,-}$. This yields

$\displaystyle\Delta I$ $\displaystyle=I_{c,+}+I_{c,-}$ (S10)
$\displaystyle=-2\eta\frac{1}{L_{1}e}\frac{\hbar}{l_{J}}\left(\langle
n\rangle\,-\frac{3}{\pi}\frac{\Phi}{\Phi_{0}}\right)\,,$ (S11)

which determines the slope $\alpha=d\Delta I/dB_{\perp}$. We note that if the
wire 2 is normal, then $n_{2}=0$, $\eta=0$ and $\Delta I$ vanishes. The
$B_{\perp}$-dependence of the critical current $I_{c,\pm}$ in such case would
merely show a monotonic decrease (without non-reciprocity) corrresponding to
suppression of the superconducting gap in Al. Experiments show a few distinct
oscillations in the asymmetric non-reciprocal part $\Delta I$ of the critical
current, see Figs. 1c and S3. We attribute the experimentally observed
vanishing amplitude of $\Delta I$ (loss of non-reciprocity) at fields higher
than $B_{\perp}\approx 750\mathrm{mT}$ to the destruction of proximity effect.
We can model this by taking $\eta$ in Eq. (S11) to be magnetic field
-dependent, detailed below.

Proximity effect is also destroyed by an in-plane field $B_{\parallel}$ along
the wire (along $x$) at roughly the same $750\mathrm{mT}$ scale, see Fig. S2a.
Since the wire 2 is proximitized in our model, we include a linear in the
field suppression of the Cooper pair density $n_{2}$ at fields lower than
those describing the superconducting gap suppression in the Al wire. This
leads to $\eta=\eta_{0}{(1-|\mathbf{B}|/B_{\mathrm{InAs,c}})}$ in Eq. S11.
Here We take $B_{\mathrm{InAs,c}}\approx 750\mathrm{mT}$ and denote
$|\mathbf{B}|=\sqrt{B_{\perp}^{2}+B_{\parallel}^{2}}$ assuming that the
suppression of proximity is isotropic (in a magnetic field parallel to
heterostructure layers). The linear suppression is taken to match with
experimental observations. In particular, a linear field-dependence is seen in
Fig. S3b where the slope $\alpha$ is plotted as a function of $B_{\parallel}$.
The measurement shows also that the switching current does not differ much
from its $B_{\perp}=0$ value (see Fig. S2c), indicating that the critical
current is determined by Al wire, as we assumed in Eq. (S9).

We thus obtain the following expression for the non-reciprocal contribution to
the critical current, plotted in Fig. 4 and Fig. S9,

$\displaystyle\Delta I(B_{\perp},B_{\parallel})$
$\displaystyle=-2\eta_{0}\frac{1}{L_{1}e}\frac{\hbar}{l_{J}}\left(\langle
n\rangle\,-\frac{3}{\pi}\frac{\Phi}{\Phi_{0}}\right)\left(1-\frac{|\mathbf{B}|}{B_{\mathrm{InAs,c}}}\right)$
(S12)
$\displaystyle\approx-2\eta_{0}\frac{1}{L_{1}e}\frac{\hbar}{l_{J}}\left(c\frac{\Phi}{\Phi_{0}}-\delta
n\sin\frac{2\pi\Phi}{\Phi_{0}}\right)\left(1-\frac{|\mathbf{B}|}{B_{\mathrm{InAs,c}}}\right)\,,$
(S13)

where $c=(1-\frac{3}{\pi})\approx 0.05$ and we assumed strong quantum
fluctuations of $n$, see discussion below Eq. (S8). The approximate period is
$B_{\perp}=\Phi_{0}(3/\pi^{2})/(l_{J}d)$, experimentally observed to be
approximately $400\mathrm{mT}$. This period indicates $500\mathrm{nm}$ for the
effective size of the vortex, given that $d=10\mathrm{nm}$.

From Eq. (S13) we obtain a zero-field slope $d\Delta I/dB_{\perp}\approx-
c_{0}\eta_{0}d/L_{1}$ where $c_{0}=(2\pi/3)\left(c-\delta n2\pi\right)$ is an
unknown numerical coefficient (since $\delta n$ is unknown). However, the
dimensionless quantity $\delta n_{q}\ll 1$ characterizes the amplitude of the
persistent current in the loop (Fig. 4) and is suppressed due to quantum phase
slips [40]. We can therefore take $c_{0}\approx 2\pi c/3\approx 0.1$. Using
values $S_{1}=150\mathrm{nm}\times 10\mathrm{nm}$, $n_{1}=18\cdot
10^{28}\text{m}^{-3}$ and $m_{1}=9.1\cdot 10^{-31}\text{kg}$ (Al electron
density and effective mass), we obtain
$d/L_{1}=\sqrt{l/\xi}S_{1}\frac{e^{2}n_{1}d}{m_{1}}\approx
3.4\mathrm{mA}/\mathrm{T}$. By comparing to the zero-field slope $d\Delta
I/dB_{\perp}\approx 1.6\mu\mathrm{A}/\mathrm{T}$ in Fig. S2b, we obtain
$\eta_{0}\approx 10^{-2}$. This is consistent with an estimate
$\eta_{0}\approx 10^{-2}$ based on the ratio of Al and InAs kinetic
inductances. We note that non-reciprocal component is proportional to the
Cooper pair density, $\Delta I\propto L_{2}^{-1}\propto n_{2}$, which is
consistent with the temperature-dependence of both quantities plotted in Fig.
S2a.

Different properties of wires 1 and 2, i.e., their asymmetry, is essential to
get non-reciprocity in our model. If the wires were identical, the wire that
switches to normal state first [in Eq. (S9)] would change upon reversing the
current direction. We note that non-reciprocity emerges even if there is no
loop ($l_{J}\to 0$) due to Josephson vortex and no phase winding, $n=0$ from
Eq. (S7), but there is nevertheless a circulating diamagnetic current
$I_{dia}$, Eq. (S6), leading to non-reciprocity, Eq. (S12), due to the assumed
Josephson coupling induced phase locking $\phi_{1}=\phi_{2}$ between the
wires.

##  IX Determining total critical current in a 2 wire model

Figure S10: Schematic of current distribution between the wires
$I_{i}=I_{0i}-(-1)^{i}I_{dia}$, $i=1,2$, and the phase difference $\Delta\phi$
as a function of an external current $I_{ext}=I_{1}+I_{2}$ for (a)
$\beta\leq\eta-\eta^{\prime}$,
(b)$\eta-\eta^{\prime}<\beta<\eta+\eta^{\prime}$ and (c)
$\beta\geq\eta+\eta^{\prime}$.

The total critical current in a two-wire model depends on the two
dimensionless parameters: the ratio of kinetic inductances
$\eta=L_{k1}/L_{k2}$ and the ratio of critical currents $\beta=I_{c2}/I_{c1}$.
(The former also determines the current distribution in the absence of
magnetic field, $\eta=I_{2}/I_{1}$.) In a magnetic field, the total critical
current will also depends on the diamagnetic current, which enters via the
dimensionless ratio $\eta^{\prime}=(\eta+1)I_{dia}/I_{c1}$. The expression of
NRC depends on the magnitude of $\beta\gtrless\eta+\eta^{\prime}$.

For $\beta\geq\eta+\eta^{\prime}$, the wire 1 turns normal first for both
directions of $B_{y}$, $I_{c}(B)=(1+\eta)(I_{c1}-I_{dia})$ and

$\Delta I=-2(1+\eta)I_{dia}$ (S14)

For $\beta\leq\eta-\eta^{\prime}$, the wire 2 turns normal first for both
directions of $B_{y}$ and

$\Delta I=+2\frac{1+\eta}{\eta}I_{dia}$ (S15)

Finally, for $\eta-\eta^{\prime}\leq\beta\leq\eta+\eta^{\prime}$, the wire 1
turns normal first for $B_{y}>0$ and the wire 2 turns normal first for
$B_{y}<0$ resulting in

$\Delta I=\frac{1+\eta}{\eta}\left[(\eta-\beta)I_{c1}+(1-\eta)I_{dia}\right]$
(S16)

The current distribution between the wires for these three cases is shown
schematically in Fig. S10

We note that the slope $d\Delta I/dB_{\perp}$ differs by a large factor
$1/\eta$ depending on which wire turns normal first. Our data is consistent
with Al (wire 1) turning normal first (at that point the whole structure is
turned normal).