Evidence of $φ$0-Josephson junction from skewed diffraction patterns in Sn-InSb nanowires

1- The authors show that the phenomenon reported does not result from fine-tuning.
2- Code and data are fully available, with datasets containing data beyond the manuscript's content.
3- The authors provide a theoretical model that mostly reproduces the measured data qualitatively.
4- The manuscript has a clear message and the story is coherent.

1- The microscopic model can only reproduce quantitatively the observed phenomenon if the spin-orbit strength is overestimated.
2- Some discrepancies between the tight-binding simulations and experimental results are not discussed.

The authors report Josephson current measurements in Sn-InSb nanowires. The results show evidence of $\phi_0$-Josephson effect. In the manuscript, the authors provide a phenomenological explanation backed up with microscopic simulations to interpret their measurements. Furthermore, the authors performed parameter sweeps to ensure that the observed effect requires no fine-tuning.

I list below a series of points that I consider relevant to be addressed before publication.

1. The so-called "phenomenological model" is not more than a harmonic decomposition of the critical current. I understand that it gives an intuitive explanation to the reader, and is also backed up with previous works suggesting that the decomposition is sound. But I do not think it is appropriate to call it a "model". If the authors are willing to name it a phenomenological model, I would expect a relation between spin-orbit field direction and the parameters in Eq. (1). Otherwise, naming it "harmonic decomposition" or something similar sounds more appropriate and requires no further changes in the manuscript.
2. I suggest, if the authors consider it appropriate, to bring Fig. 2 to the beginning of the manuscript to present the device layout early on.
3. The authors mention that the tight-binding model shows $\gamma=0$ in the absence of Zeeman field. However, figure 2a shows $\gamma\neq0$ for $B_x=0$. This result does not match the tight-binding results shown in the same figure (panel 2b). Can the authors provide an explanation for this discrepancy?
4. The authors say:
"Orbital effect only ($\alpha = 0,~A\neq 0$) yields a similar structure (see $B_x = 80\mathrm{mT}$), limited in magnitude and field. We believe this is a simulation artifact that appears when the external field is perpendicular to the line connecting the center of the wire and the center of the shell. However, there is no explanation to justify that the results are a simulation artifact."
Can the authors expand on this statement and/or provide numerical evidence that the statement is sound?
5. Since the authors mention that the simulated device has the same geometry as the measured nanowires and all the parameters are chosen to match the experiments, why is it necessary to overestimate the spin-orbit coupling? Could that hint that SOC is not the main cause of the phenomenon?
6. In both the experiments and the simulations it's possible to observe finite $\gamma$ even when $\mathbf{B} \perp \hat{x}$. I understand that it is likely hard to resolve the direction of the spin-orbit field in the experiments. But I am wondering why the simulations show a similar feature.

The following comments are about the Supplementary Materials.

7. In Sec. V.B the authors report hysteresis as a function of bias voltage. Can the authors provide an explanation for the presence of this hysteresis?
8. In Sec. VI, the authors say that the gauge is fixed so that the system "is [translationallly?] invariant along the $x$-direction". However the gauge choice explicitly depends on $x$, so I don't follow the statement. Do you rather mean that the field is perpendicular to $\hat{x}$?
9. In Sec. VI, the authors say that "kwant.continuum.discretize cannot handle the systems with lower symmetry", however the [tutorial](https://kwant-project.org/doc/1/tutorial/discretize) shows an example with no translational symmetry. It is unclear to me what the authors mean by this statement.

1- Explain the discrepancies between the two panels in Fig. 2.
2- Avoid calling Eq. (1) a phenomenological model, or relate the parameters in the equation with system parameters.
3- Appropriately address the questions in the referees' reports.

The authors analyze the critical current behavior for a hybrid Sn/InSb
junction in a magnetic field. They point out an asymmetry in the forward
and reverse current which they associate with phi0 behavior of the
junction current phase relation. Their claim is supported by some model
calculations including spin-orbit interactions.

In my opinion, the results presented in this manuscript are sound.
However, I have some concerns with respect to the authors interpretation
and the way that the results are presented.

First of all, while the authors focus on phi0 behavior as the cause of
the asymmetric critical current patterns in a magnetic field,
it should be stressed that an anomalous phase shift is by itself not
enough to produce such asymmetries. Rather than that, the asymmetric
patterns require a diode effect, which is not necessarily linked to the
phi0 behavior. While this is implicit in the discussion around Eq. (2),
I find that the title and the abstract are misleading as a pure phi0
effect can only be detected through phase biased measurements.

On the other hand, phi0 behavior in these type of junctions in a magnetic
field is not surprising and have been reported in previous works like
Ref. 34 in the manuscript (see also https://arxiv.org/abs/2208.11198 for
more recent experiments based on microwave spectroscopy in phase biased
InAs junctions). In contrast, the superconducting diode effect have been
observed mainly in van der Waals junctions and, to my knowledge, it is
at present not fully understood.

For these reasons I believe that the manuscript requires a thorough
revision in order to clarify the difference with previous works in which
phi0 behavior has been reported.

In addition, I have a number of comments that should be addressed before
publication:

- In device A the junction is placed closer to the left Au lead. For this
reason the Sn shell on the left could be more affected by inverse
proximity effect than the right one. Although it is not likely that this
could have an influence on the supercurrent asymmetry it could be relevant
for the junction transport properties and should be commented.

- The supercurrent is measured in a two terminal configuration. It would
be worth that the authors comment on the method used to eliminate the
possible effect of any series resistance and its size.

- Regarding the tight-binding model calculations, the size of the spin
orbit parameter which is necessary to get an effect of the same order
as in experiments is extremely large compared to existing estimates. I
wonder whether this could be pointing out to some missing ingredient in
the modeling. It would be convenient that the authors give more details
on these calculations.

- On the other hand I have some concerns regarding the paper organization.
For instance, the device description in Figure 2 is then repeated in the
description of Figure 4. I also don't understand the importance given
to the "phenomenological model" which is just a mathematical expression
of a current phase relation with two displaced harmonics.