Link between supercurrent diode and anomalous Josephson effect revealed by gate-controlled interferometry

In Josephson diodes the asymmetry between positive and negative current branch of the current-phase relation leads to a polarity-dependent critical current and Josephson inductance. The supercurrent nonreciprocity can be described as a consequence of the anomalous Josephson effect —a φ0-shift of the current-phase relation— in multichannel ballistic junctions with strong spin-orbit interaction. In this work, we simultaneously investigate φ0-shift and supercurrent diode efficiency on the same Josephson junction by means of a superconducting quantum interferometer. By electrostatic gating, we reveal a direct link between φ0-shift and diode effect. Our findings show that spin-orbit interaction in combination with a Zeeman field plays an important role in determining the magnetochiral anisotropy and the supercurrent diode effect.

The same symmetries need to be broken in order to observe the supercurrent diode effect (SDE), namely, the dependence of the critical current on the bias polarity.This effect can be trivially obtained, e.g., in asymmetric superconducting quantum interference devices (SQUIDs) [38] or, more generally, when in a film an inhomogeneous supercurrent distribution is coupled to a flux.Recently, it was shown [17-19, 22-27, 30, 31] that supercurrent rectification can as well emerge as an intrinsic feature of homogeneous quasi-2D systems subjected to a Zeeman field.Such nontrivial SDE is a new precious probe of the condensate physics (in films) [10] and of Andreev bound * nicola.paradiso@physik.uni-regensburg.destates (ABSs) in Josephson junctions [27,30], including possible topological properties [39][40][41].
Early observations on Josephson diodes were interpreted in terms of  0 -shift in ballistic systems with skewed CPR [18,[22][23][24]56].Within this picture, the SDE ultimately originates (as it does the  0 -shift) from SOI.An alternative model, proposed by Banerjee et al. [30] based on the theory of Ref. [50], explains the same effect in terms of a purely orbital mechanism.To date, it is not clear yet to which extent the two mechanisms (namely, the SOI-based and the purely orbital mechanism) contribute to the supercurrent rectification observed in experiments.
In this work, we make use of an asymmetric SQUID with mutually orthogonal junctions to directly measure both the anomalous  0 -shift and the SDE on the same junction.By gating, we can electrostatically control both effects and highlight their relation.Finally, by measuring the temperature dependence of the  0 -shift and of the diode efficiency we demonstrate that the former is a necessary but not sufficient condition for the SDE.This latter requires in fact the presence of higher harmonics in the CPR, which are quickly suppressed by increasing the temperature.We comment on our results in light of alternative models proposed in the literature and compare the temperature dependence of  0 and the SDE to the predictions of a minimal theoretical model.

II. DEVICE DESCRIPTION AND EXPERIMENTAL RESULTS
Figure 1a shows a scheme of our SQUID.The device is fabricated starting from a molecular beam epitaxy-grown heterostructure featuring an InGaAs/InAs/InGaAs quantum well, capped by a 5 nm-thick epitaxial Al film [18,[58][59][60].The quantum well hosts a 2D electron gas (2DEG) with a proximityinduced superconducting gap inherited from the Al film.By deep wet etching, we define an asymmetric SQUID loop.The actual geometry is shown in the false-color scanning electron microscopy image in Fig. 1b, where the turquoise areas indicate the pristine Al/InGaAs/InAs/InGaAs regions, while the gray areas refer to deeply etched (insulating) regions.To obtain the two normal (N) weak links, we selectively etch the Al film (yellow areas in Fig. 1b).The reference Josephson junction 1 (JJ1) is 28 µm-wide and 120 nm-long, whereas the Josephson junction 2 (JJ2) is 2.7 µm-wide and 100 nm-long.Finally, a gate is fabricated on top of JJ2, which allows us to control the electron density in the N-link and thus the critical current  ,2 of this junction.The two junctions are mutually perpendicular, so that an in-plane magnetic field ì  ip parallel to the short junction, i.e., along ŷ, (see Fig. 1a,b) will induce magnetochiral effects [18] in the short junction only, and not in the reference junction.Here, we take as positive ẑ direction that perpendicular to the 2DEG and directed from the substrate towards the Al (see Fig. 1a), which corresponds to the direction opposite to the built-in electric field in the quantum well [18,61] which provides SOI in the 2DEG [18,61].
We measure differential resistance in a 4-terminal geometry as a function of DC current.In what follows, we indicate as   the (measured) SQUID critical current (see Methods) and as  , ( = 1, 2) the (deduced) critical current in junction .When needed, we use the superscript + (−) to indicate positive (negative) current from source to drain (see Fig. 1a), i.e., supercurrent in the positive (negative)  direction in JJ2.For the in-plane field, we interchangeably use either Cartesian components, or magnitude and angle parametrization, i.e. ì  ip =   x +   ŷ =  ip (cos , sin ), see cartesian axes in Fig. 1a.
Figure 1c shows the color plot of the differential resistance versus out-of-plane field   and DC current , measured with an applied field   = −100 mT at  = 40 mK.The fast oscillations have period 1.8 µT, corresponding to a flux quantum Φ 0 = ℎ/2 applied to the loop.Such oscillations are superimposed to the Fraunhofer pattern of the reference junction, whose central and first side lobes are visible.The plot displays an evident asymmetry around the   = 0 and  = 0 axes, while it is approximately point-inversion-symmetric around the origin, namely   (  ) ≈ −  (−  ).As discussed in the Supplementary Information, such asymmetry is due to screening effects, which arise when there are current-dependent corrections to the fundamental SQUID relation where   it the gauge-invariant phase drop at the -th junction, Φ 0 is the flux quantum and Φ the flux through the loop.The corrections originate either from the additional flux induced by the SQUID current itself, or by the phase drop accumulated along the loop arms [38,62].The former correction is proportional to the geometric loop inductance, while the latter is proportional to the kinetic inductance of the SQUID arms [63].As discussed in the Methods, in our sample, screening is mostly dominated by the large kinetic inductance of the thin Al film, whose sheet inductance is  □ ∼ 30 pH.The geometric loop inductance is comparatively small, see Supplementary Information.As shall be discussed below, screening effects hinder the measurement of both the absolute value of the anomalous shift  0 and that of the diode efficiency.Indeed, the determination of the absolute  0 is challenging even in SQUID devices with low screening and reference devices [32,36].One of the difficulties is the fact that an accurate determination of  0 requires a reproducible   control on the microtesla scale, while an in-plane field of the order of tens or hundreds of millitesla is swept.The determination of the absolute  0 is usually difficult in systems with large kinetic inductance.Similar to Refs.[33,34,36], we shall measure the relative shift of the CPR with respect to that for large negative gate voltage.Figure 1d shows SQUID oscillations measured for different gate voltages   .The measurement is performed with an in-plane field  ip = 100 mT applied perpendicular to JJ2 ( = −90 • ) at  = 40 mK.The oscillations are taken near the maximum of the tilted Fraunhofer pattern, indicated by the upper arrow in Fig. 1c.The black curve (  = −2.5 V) refers to a completely pinched-off JJ2 ( ,2 = 0), and serves as zero-current baseline.The first curve for which oscillations are clearly visible is that for   = −1.5 V. We shall label as  0 (  ) the crossing with positive slope of each   (  ) curve with the baseline   (  = −2.5 V), see Fig. 1d.As discussed in the Supplementary Information, at these crossings the supercurrent  2 in JJ2 vanishes, therefore its gauge invariant phase difference is, by definition, the anomalous shift  0 .We consider variations of  0 with respect to the reference voltage where  loop = 1150 µm 2 is the loop area.
Figures 1e and 1f show the results of the same measurements after two subsequent -90 • sample rotation, namely, for  = 180 • and  = 90 • , respectively.From Fig. 1e we deduce that for ì  ip ∥ ì , the curves crosses the baseline with positive slope nearly at the same   (i.e.,  0 (  ) is constant).Instead, for ì  ip ⊥ ì ,  0 (  ) monotonically increases (decreases) with   for negative (positive) sign of ê • ( ì  ip × ì ).This is a clear signature of the magnetochiral nature of the anomalous Josephson effect [18].
The variation of Δ 0 (  ) for different ì  ip orientations (i.e., for different  with | ì  ip | = 100 mT) is plotted in Fig. 1g.We stress that, since we subtract  0 (  = -1.5V)(as in the definition of Δ 0 ), what is important in Fig. 1g is the monotonic increase or decrease of Δ 0 with the gate voltage.The graph clearly shows the proportionality of Δ 0 to [18] − ê • ( ì  ip × ì ), as expected by SOI-based models for the anomalous Josephson effect [12].For all curves, the magnitude of |Δ 0 | increases with   .The monotonic increase is expected, since in this type of InGaAs/InAs/InGaAs quantum wells, a positive gate voltage increases the built-in electric field [61] responsible for the Rashba SOI [18,61].
The main goal of our experiments is to establish a relation between the anomalous  0 -shift and intrinsic SDE by measuring both phenomena on the same junction.For this purpose, we investigate SQUID oscillations for both current bias polarities in order to deduce both the positive ( + ,2 ) and the negative ( − ,2 ) critical current of JJ2. Figure 2a shows the SQUID interference pattern measured in the vicinity of   = 0 for   = −50 mT at   = 1.0 V (red), together with the reference curve at   = −2.5 V (black, where  ,2 = 0 and SQUID oscillations vanish).Figure 2b shows several  +  (  ) curves for different gate voltages   , measured at   = −50 mT and  = 40 mK, as in Fig. 2a.We focus on one particular oscillation highlighted by the upper arrow in Fig. 2a.The corresponding  −  curves for the opposite   range (lower arrow in Fig. 2a) are shown in panel c.We use the baseline curve in Fig. 2a (black,   = −2.5 V) as a reference to extract  + ,2 from data in Fig. 2b and  − ,2 from Fig. 2c.For each   ,  + ,2 corresponds to the maximum in   (see red arrow in Fig. 2b) of the difference  2e for different values of   in the different subpanels (i-v).As explained in the Supplementary Information, owing to SQUID screening effects  is affected by trivial offset, which means that data in Figs.2e might be subjected to a spurious vertical shift.Nevertheless, the variation of  when   is swept remains unaffected by screening artifacts.Since the modulation of  with   is roughly proportional to  itself, we use /  as a measure of .In particular, we expect that the slope /  must display the same magnetochiral behavior as  itself.Data in Fig. 2e(i-v) show that /  is linear in   , similar to the rectification efficiency  reported in single Josephson junctions from similar materials [18,22,24,25].The complete overview of the   -dependence of /  is shown in Fig. 2f.As in previous reports on Josephson diodes, the rectification efficiency is linear in   only up to a certain threshold (here at |  | = 50 mT): after that, a clear suppression is observed [27].
The results shown so far provide strong evidence of the link between anomalous phase shift and SDE.Both effects are linear in   .Both Δ 0 and  can be modulated by a gate voltage in the accessible range   ∈[-1.5 V, 1.0 V].This is consistent with the SOI-based model [10,18,24,53,56] of the SDE, where the supercurrent rectification is accompanied by the anomalous Josephson shift  0 .The  0 -shift is a necessary but not sufficient condition for the supercurrent nonreciprocity.To break the symmetry between positive and negative part of the CPR, multiple ballistic channels are needed [14,27].In finite-width junctions with parabolic dispersion, each channel has a different  0, -shift, owing to the different Fermi velocity  , : in fact, for a single ballistic channel,  0, ∝  −2 , [12].The total CPR is the sum of all single-channel CPR contributions   ( +  0, ).The sum of skewed CPRs with different  0, -shift leads to an asymmetric total CPR, whose positive and negative branch are different.The skewness (i.e., the content of higher harmonics) of the individual CPRs is crucial, since otherwise the sum of sinusoidal CPRs would always lead to a sinusoidal -i.e., reciprocal-CPR.The relation between  and  0 is evident from the comparison between Fig. 1g and Fig. 2e.On the other hand, since the SDE also relies on the presence of higher harmonics in the CPR [18], we expect that  (and thus /  ) will be highly sensitive to the junction transparency and temperature, as opposed to the anomalous shift  0 .
Figures 3a and 3b respectively show the temperature dependence of  0 and  as obtained from a simple theoretical model (described in Methods and in Refs.[27,53]) for a Josephson junction in the short-ballistic limit with Zeeman interaction.The junction separates two semi-infinite superconducting 2DEGs with zero-temperature superconducting gap Δ * (0) ≈ 130 μeV [60], and critical temperature  c ≈ 2 K.We note that the anomalous shift  0 is nearly -independent, whereas the supercurrent rectification is strongly suppressed already for  > 100 mK.
In Fig. 3c and 3d we show the measured temperature dependence of Δ 0 and that of /  , which we take as measures of  0 () and (), respectively.Δ 0 () is measured at   = −100 mT, ( ip = 100 mT,  = −90 • ), where the field magnitude is set large enough to limit screening effects to an acceptable level.Instead, /  () is measured at   = 50 mT, where the SDE is maximal, see Fig. 2f. Figure 3c shows that Δ 0 is temperature-independent within the experimental accuracy.In contrast, the (gate modulation of the) supercurrent rectification is clearly suppressed already at temperatures well below   , as shown in Fig. 3d.Both observations match the corresponding theory predictions.
A comment is in order about the sign and magnitude of the effects.Both our experimental data and analytical model show that if the Rashba SOI-inducing electric field is directed along − ẑ (as in Ref. [61] and in the Supplementary Information of Ref. [18]), ì  ip along + ŷ, and the positive current bias along + x, then  0 < 0 [where  ( 0 ) = 0,    ( 0 ) > 0] and  > 0. For ease of comparison, all main graphs are plotted up to  = 0.8 K, the highest temperature for which Δ 0 and  0 /  was measurable.
Instead, the magnitude of  0 and  predicted by ballistic theory [12] is smaller than the one measured in our and in other experiments in the literature [32,33].A possible explanation for this discrepancy could be disorder in and near the junction [32], since diffusive models predict a much larger  0 .Another possibility is the enhancement of SOI due to interaction of quantum well electrons with the image charges that are formed in a nearby Al gate.The nontrivial property of the induced image-potential that depends on the electron density of 2DEG gives a feedback on SOI that superimposes with the innate Rashba SOI of the quantum well without metallic gate as demonstrated in Refs.[64][65][66].

III. DISCUSSION
The main goal of our study is to elucidate the physical mechanism behind the intrinsic SDE in single, homogeneous Josephson junctions.The effect has been so far explained by two different models: one [14,18,56] is based on the combination of Rashba SOI plus Zeeman interaction (due to an external in-plane field or exchange interaction); the other is a purely orbital mechanism [30,50] based on the finite Cooper pair momentum induced in the superconducting leads by the flux associated to the in-plane field [30].This flux is finite if the parent superconducting film and the 2DEG are spatially separated.
The main difference between the two pictures is the expected dependence on   .Such dependence naturally emerges since   affects the band alignment and thus the Rashba coefficient  R .Both  R and the electron density  critically affect  0 , which, in turn, determines  in multichannel systems.In contrast, the orbital mechanism [30] hardly depends on the gate voltage [67].As discussed in the Supplementary Information, the gate voltage also affects the magnetochiral anisotropy for the inductance [18], an effect that is strictly related to the supercurrent rectification [27].The observed strong gate dependence of  0 and  indicates that the Rashba-based mechanism must certainly play an important role in the SDE.On the other hand, the orbital mechanism cannot be ruled out by our observations: it could still coexist with the spin-orbit-based mechanism.
Finally, we would like to stress that, even though we make use of a SQUID to link  to  0 , our point does not concern the (trivial and long known [38]) SDE of the asymmetric SQUID as a whole.Our focus is exclusively on the intrinsic SDE in a single, homogeneous junction (JJ2).
In conclusion, we have shown the coexistence of anomalous Josephson effect and supercurrent rectification by measuring both effects on the same Josephson junction embedded in a SQUID.The observed gate voltage and temperature dependence are compatible with a spin-orbit based picture where supercurrent rectification arises in multichannel junctions with anomalous shift  0 and skewed current-phase relation.
Josephson diodes based on  0 -junctions are important for both fundamental research and applications.They are novel and powerful probes of symmetry breaking in 2D superconductors [21,49,68] and possible probes of topological phase transitions [39].A recent proposal [69] suggested that the anomalous Josephson effect might be used in multiterminal junctions to obtain compact nonreciprocal devices as, e.g., circulators for rf-applications [70].

A. Experimental methods
The heterostructure is grown by molecular beam epitaxy.The full layer sequence is reported in the Supplementary Information.The most relevant layers are the topmost ones, namely, the nominally 5 nm-thick Al film at the sample surface, a 10 nm thick In 0.75 Ga 0.25 As layer acting as a barrier, a 7 nm InAs layer hosting the 2DEG, followed by another In 0.75 Ga 0.25 As barrier of thickness 4 nm.Structures are defined by electron beam lithography.The selective etching of Al is performed by wet chemical etching using Transene D. Deep etching processes (where the 2DEG is removed altogether) is performed using a phosphoric acid-based solution.
Transport measurements are performed in a 4-point configuration using standard lock-in techniques.To determine the SQUID critical current  +  and  −  as defined in the text we take / = 6 Ω as a threshold.
To determine the correct offset for   we look at the symmetry of the plot of  = / versus  and   , see e.g., Fig. 1c.Since (,   ) ≈ (−, −  ), the center of inversion symmetry of the plot allows us to determine the applied out-of-plane field which corresponds to an effective   = 0.

B. Theoretical methods
Our theoretical model, initially developed in Refs.[27,53], describes the experimentally relevant system in terms of a short S-N-S Josephson junction that couples two semi-infinite -wave superconducting (S) regions with inherently strong Rashba SOI through a thin delta-like normal-conducting (N) link.Nontrivial solutions of the 2D Bogoliubov-de Gennes equation [71] Ĥ with the single-electron Hamiltonian determine the energies  and wave functions Ψ(, ) of the Andreev bound states [72], which are at the heart of the coherent Cooper-pair supercurrent transport along the x-direction; Δ() corresponds to the -wave superconducting pairing potential that we approximate by where Δ * () = Δ * (0) tanh(1.74√︁  c / − 1) is the temperature-dependent proximity-induced superconducting gap [from the experimental data, the induced gap at zero temperature was estimated as Δ * (0) ≈ 130 μeV and the critical temperature as  c ≈ 2 K] and  indicates the phase difference between the two superconducting regions.The Rashba SOI that is present throughout the whole system is parameterized by  R ,  0 and  Z represent the scalar (spin-independent) and Zeeman (spin-dependent) potentials inside the delta-like N link of thickness -the magnetic field causing the Zeeman splitting is thereby aligned perpendicular to the current direction (i.e., along ŷ)-,  is the Fermi energy,  the (effective) quasiparticle mass, and σ0 and σ refer to the 2 × 2 identity and th Pauli spin matrix, respectively.
After determining the Andreev-state energies  () as a function of the superconducting phase difference , we apply the quantum-mechanical current operator to the corresponding bound-state wave functions inside the N link to compute in the first step the Josephson CPRs  () and obtain in the second step the direction-dependent critical currents necessary to quantify the SDE.In the simultaneous presence of SOI and Zeeman interaction, the bound-state energies depend on  through  () = Δ * ()  (), where the generic function  () is no longer antisymmetric with respect to , i.e.,  (−) ≠  (), reflecting the broken space-inversion and time-reversal symmetries, and the therefrom resulting nontrivial  0 -phase shifts.Note that the only impact of temperature on  () is an effective rescaling (i.e., suppression with increasing temperature) of the superconducting-gap amplitude Δ * () according to Δ * () = Δ * (0) tanh(1.74√︁  c / − 1), whereas the qualitative shape of  () is not altered by temperature.The total Josephson current is then given by where the sum ensures to account for the current contributions of all bound states (i.e., from all transverse channels of the junction) and  B indicates the Boltzmann constant.The current at zero temperature can, in the simplest case, be extracted from the thermodynamic relation [73]   (); with the positive elementary charge .The major temperature effect on the Josephson current originates therefore from the suppression of the higher-harmonic contributions in Eq. ( 6) due to the tanh term in Eq. ( 5).

XI. MODELING OF INDUCTIVE SCREENING EFFECTS IN THE ASYMMETRIC SQUID
The obtained critical currents of the SQUID device are strongly affected by the large kinetic inductance of the aluminum electrodes.A minimal circuit model of our device, which includes the inductance of the electrodes of JJ1 and the loop, is show in Figure S2a.The reference junction JJ1 is modeled as a network of  JJs in parallel, each with critical current  ,1 /.Setting  = 20 is sufficient to model the central lobe of the Fraunhofer interference pattern of JJ1.The effective area of JJ1 is  ref , while the area of the large loop is  loop .The positive and negative critical currents for a given out-of-plane field   can be calculated as a function of the phase difference  JJ2 over JJ2.Given  JJ2 the phase difference  JJ1,N over the rightmost junction in JJ1 is The phase differences  JJ1, are found by iteration: where   =  +1 +  ,1 / sin( JJ1,+1 ) is the current flowing through the inductor between junctions  and  + 1.The positive and negative critical currents are then found as the extremal values of  0 ( JJ2 ).In FiguresS2b-d we model the SQUID critical current obtained for a in-plane-field |  | = 100 mT.A good match to the data can be found with the following parameters: The amplitude of the SQUID oscillations at   = 1 V,  = −100 mT and  = 40 mK is matched by setting  ,2 = 400 nA. Figure S2b shows the SQUID critical current for different values of  ,2 near the maximum of the SQUID critical current.Importantly, the crossing point with the reference line ( ,2 = 0) is the same for all values of  ,2 , showing that screening has no effect on our evaluation of Δ 0 .
Figure S2c shows the results of SQUID oscillation measurements performed for   = 1.0 V (green) and   = −2.5 V (purple), at   = −100 mT and  = 40 mK.A comparison with the results of our calculations in Fig. S2d makes it evident that our simple model described above correctly captures the behavior of the SQUID.We want to stress that our asymmetric SQUID displays a trivial supercurrent diode behavior (in Fig. S2c,d in general  +  (  ) ≠  −  (  )).This is a long known feature of asymmetric SQUIDs and it is not the focus of our work.Instead, our focus is on the physics of a single, homogeneous Josephson junction (JJ2).
From the analysis of the inductive screening effects discussed above, we can conclude that: • Owing to screening effects, the SQUID oscillations do not reproduce the CPR of JJ2.However, for positive bias, one can still deduce  + ,2 from the maximum of the oscillations, since these two quantities are proportional, see Fig. S2d.Instead, for positive bias, one cannot deduce  − ,2 from the minima of the SQUID oscillations.The correct way to deduce  − ,2 is to look at oscillations for negative current bias. − ,2 is proportional to the minima of the SQUID oscillations for negative bias, see Fig. 2c of the main text.
• As it is clearly visible in Figures S2c,d |).We note that the gate slope /  is nevertheless a well defined quantity and not affected by the screening effects.
• As a consequence of the   -dependence of the amplitude of the SQUID oscillations, the correct way to compare an oscillation with positive bias with one with negative bias is to consider oscillations with opposite   , namely oscillations which are inversion-symmetric around the origin of the   (  ) graph, see arrows in Fig. 2a of the main text.Indeed, by the global time-reversal symmetry  +  (  ) = − −  (−  ) in absence of in-plane field.Taking precisely opposite oscillations, as described in the previous point, should make screening effect irrelevant, so that, in principle, the observation of finite  for finite  ip should be considered as a real effect and not an artifact of screening.However, the combination of  0 -shift and the strong dependence of the SQUID oscillations on   makes it possible to measure (following the aforementioned procedure) a finite  even in absence of a genuine SDE in JJ2.In fact, owing to the finite  0 , opposite SQUID oscillations do not occur anymore at exactly opposite   , and the strong   dependence of the SQUID oscillations can render the opposite oscillation amplitudes not exactly equal even in absence of intrinsic SDE.The effect is very small, still we maintained a conservative approach and decided to look at /  instead of  itself.Also, we can exclude that the gate dependence of  is a simple artifact of that of  0 (due to the mechanism just described) for the following reasons: (i) from our calculations the measured  is too large to be entirely generated by the mechanism just described ( 0 plus   -dependent oscillations).(ii) The gate dependence of  0 and  are different: the former is nonlinear, the latter is linear (compare Fig. 1g and Fig. 2e of the main text).(iii) The temperature dependence of Δ 0 and that of /  is dramatically different, see Fig. 3 of the main text.
• From Fig. S2b we can conclude that inductive screening does not affect the value of   at which  2 = 0 (the curves all cross the baseline at the same point).Clearly, for positive bias, it is important to only consider the crossing with the baseline with positive slope, which is the significant one.This explains our operational definition of  0 , which is important to determine  0 .
• The inductive screening effects depend on the kinetic inductance of the Al film and on the magnitude of the supercurrent flowing the in the SQUID arms.By increasing  ip in the range under study, the decrease of the critical current   is more important than the increase of the kinetic inductance of the arms.Therefore, the larger  ip , the smaller the impact of screening.Since  0 measurements are more sensitive to inductive screening, we were able to extract Δ 0 only for  ip ≥ 100 mT.

XII. GATE DEPENDENCE OF THE MAGNETOCHIRAL ANISOTROPY FOR THE INDUCTANCE
In this section we comment about the gate dependence of the magnetochiral anisotropy (MCA) of the inductance, which we measured in our previous work, Ref. [18].In that work, we established an analogy between the known MCA for the resistance a b c d (e.g. in the fluctuation regime of a noncentrosymmetric superconductor) and MCA for the inductance.This latter was the new quantity introduced in Ref. [18], where we showed that the Josephson inductance can be written as which holds for noncentrosymmetric conductors with nonreciprocal resistance, for which  is defined as 2 ′ 0 /( 0    ), with  ′ 0 ≡   | =0 .It must be kept in mind that both the MCA coefficient for the resistance () and that for the inductance (  ) are dimensional quantities which scale with the inverse of the sample width in quasi 2D conductors.In Ref. [18], we have plotted in Fig. 2d the gate dependence   .At first glance, it might appear that no dependence is observed (the curves in Fig. 2d of Ref. [18] fall on top of each other).However, it must be kept in mind that   trivially depends on the number of channels  (or, equivalently, on the width).Since  ∝  and  0 ∝  −1 , then   must scale as 1/.Therefore, in Ref. [18] the fact that   appears constant is due to the fact that the expected increase of the specific magnetochiral character (considered "per channel") is compensated by the decrease due to the trivial increase of  (decrease of  −1 ).
The best way to represent the gate-induced enhancement of the MCA for the inductance (which mirrors the increase in | 0 | and  with   demonstrated in the present article) is to represent the MCA with a quantity which does not depend on .To this end, we plot in Fig. S3a the   dependence of the quantity Φ 0 | ′ 0 |/ 2 0 for the same data set of Fig. 2d in Ref. [18].Data refer to  ip = 100 mT and  = 90 • (purple) and  = 270 • (green).We clearly observe an increase in the magnetochiral character with   .The increase is relatively small due to the fact that in that sample the gate had a much weaker effect compared to the sample studied in the present article.To give an idea of this, in Fig. S3b we plot  0 as a function of   , showing that sweeping   from -1.5 to 0 V corresponds to a change in  0 of less than 20%.

XIII. ANGLE DEPENDENCE OF THE ANOMALOUS PHASE SHIFT
Figure S4a shows the dependence of the anomalous phase difference Δ 0 (  = 1.5 V,  ip = 100 mT) on the rotation angle  defined in the main text.The observed behaviour is in good agreement with the expected relation  0 ∼ sin().In Fig. S4b we plot Δ 0 as a function of the   component of the in-plane field (  :=    sin()), showing a good agreement with the expected linear behaviour.

XIV. SUPPLEMENTAL DATA ON TEMPERATURE DEPENDENCE
Figure S5a shows the temperature dependence of the differential resistance / of the SQUID for zero DC bias current and zero magnetic field.The positive and negative critical currents  ± ,2 of JJ2 corresponding to Fig. 3c and d of the main text are shown in Fig. S5b.At  = 750 mK the critical current is reduced by ∼ 50 %.Fig. S5c and d show (  ) for the temperatures  = 40 mK and  = 750 mK (corresponding to the first and last data points in Fig. 3d of the main text).At  = 750 mK we observe larger statistical errors in the determination of the critical currents, leading to the larger error bars.By increasing the temperature, the superconducting-to-normal transition in the differential resistance becomes less sharp, leading to the increased noise in the determination of critical currents, and thereby the diode efficiency.

FIG. 1 .
FIG.1.Asymmetric SQUID device with reference junction and gate-controllable  0 junction.a, (Left) Schematic illustration of the device.The SQUID consists of a large reference junction (JJ1) and of a small junction (JJ2) which is coupled to a gate.By applying a magnetic field along ŷ, a  0 -shift is induced in the CPR of the latter junction, which can be controlled by electrostatic gating.(Right) Scheme of the topmost layers of the heterostructure.The black arrow indicates the direction of the positive current bias .b, False-color scanning electron microscopy image of the device taken before gate patterning.The pristine superconducting Al/InAs leads are highlighted in turquoise, the areas where Al is selectively etched in yellow (including the weak links, highlighted by the black arrows).The remaining parts in gray correspond to deeply etched regions, where both the Al film and the topmost semiconducting layers are etched.c, The color plot shows the SQUID differential resistance versus out-of-plane field   and current , for  = −90 • and  ip = 100 mT (i.e.,   = 0,   = −100 mT) at  = 40 mK.The white arrow indicates where anomalous phase shifts were measured (see panels d-f).d, SQUID critical current   as a function of   , for  and  ip as in c.The different curves refer to different gate voltages   .We define  0 (  ) as the crossing of each curve with the   = −2.5 V reference curve (black).e and f, Corresponding measurements for the same  ip = 100 mT but, respectively,  = 180 • and  = 90 • .g, Plot of Δ 0 (  ) ≡ 2 [ 0 (  ) −  0 (  = −1.5 V)]/Φ 0 , for different ì  ip orientations, i.e., for different .Here,  is the loop area and Φ 0 the flux quantum.

FIG. 3 .
FIG. 3. Temperature dependence of  0 and : theory and experiment.a, Computed temperature dependence of the (sign changed) anomalous phase shift  0 for a Zeeman parameter of  Z = −0.40(which would correspond to   = −100 mT for -factor || = 12).We assume   = 2.0 K as in the experiments.Inset: full-range  0 () graph plotted for  up to  =   .b, Computed temperature dependence of the supercurrent diode rectification efficiency − for the same parameters as in a. Inset: full-range −() graph up to  =   .c, Measured Δ 0 (as defined in the text) at   = 1.0 V and   = −100 mT, plotted as a function of temperature.d, Temperature dependence of  0 /  (as defined in the text) at   = 50 mT.For ease of comparison, all main graphs are plotted up to  = 0.8 K, the highest temperature for which Δ 0 and  0 /  was measurable.
FIG. S1. a, Layer stack of the used wafer.b-e, Transport properties of the quantum well measured with a top gated Hall bar at temperature  = 1.5 K.The inset of panel e shows the Fermi level   versus .The range in gate voltage corresponding to the -range displayed in panels b-e is [-3.5 V,-0.2 V]. f, Amplitude of Shubnikov-de Haas oscillations for temperatures between 1.4 K and 20 K. g () Characteristic of a Hall bar device with the pristine aluminum film.
48 A,  ref = 70 (µm) 2 ,  ref = 0.8 nH, and  loop = 1.5 nH.From the value of  loop we determine the sheet inductance of the aluminum film  □ =  loop / □,loop = 1.5 nH/46 ∼ 30 pH.The kinetic inductance of the loop largely exceeds the geometric inductance which we estimate as  loop,geo ≲ 100 pH.
FIG. S2. a, Minimal circuit model of the SQUID device.b,d, Simulated critical currents of the SQUID with parameters as described in the text.c, experimental values of positive and negative critical currents for  ip = 100 mT and  = −90 • .

T
FIG. S5. a, / () of the SQUID device at zero DC bias and zero magnetic field.b, Critical current  ± ,2 (  ) at   = −50 mT for different temperatures.c and d, Diode efficiency (  ) for  = 40 mK and  = 750 mK.
, the amplitudes of the fast SQUID oscillations are not constant.This is caused by a coupling of the loop current  JJ2 to the flux of JJ1.The flux in JJ1 is changed by Φ ∼  ref  JJ2 , modulating the critical current of JJ1.For the measurement of critical currents, demonstrated in Figure2of the main text, we find that the critical currents for different gate voltages are only proportional to  ,2 (  ), with the proportionality constant depending on the chosen SQUID oscillation (i.e., on the exact   value).As positive and negative critical currents need to be measured at opposite current bias (see previous points), the slightly different scale factors result in offsets for the extracted diode efficiency  = 2( +  − | −  |)/( +  + | −