Proximity-induced gapless superconductivity in two-dimensional Rashba semiconductor in magnetic field

The main strength is the clear and thorough investigation of a theoretically important and experimentally relevant problem.
The paper is very well written and presents a consistent study of the gapless state tied to the Bogoliubov Fermi surfaces as well as the superfluid density in the spin-orbit coupled proximitized two-dimensional system subject to a parallel magnetic field.

Dear editor,
Please find the report below.

Authors investigate the density of states and the superfluid density of the proximitized two-dimensional electron gas (2DEG) with Rashba spin-orbit coupling in the presence of the in-plane magnetic field. The boundary between the systems is assumed to be fully transparent. The authors carefully investigated the dependence of the evolution of the gapless superconductivity with the disorder in the 2DEG and the strength of the magnetic field.
The order parameter is fixed by the superconducting substrate assumed to be in a dirty limit.
The gap in the substrate is determined by numerically solving the Eilenberger equations in conjunction with the self-consistency equation. The density of states is obtained by solving the Gorkov equations using the iteration technique.
The disorder in the 2DEG is shown to stabilize the gapless state which is shown to emerge from the Bogliubov Fermi surfaces appearing in the clean system as the Zeeman splitting reaches the superconducting gap. At elevated magnetic fields the negative superfluid density is interpreted as indicative of an instability. Authors list few options as of how such instability could have been resolved.
Finally the authors perform the detailed six-parameter fit of their theoretical results to the existing experimental data. The fit is shown to be very good.

I would like to recommend this high quality work for publication.
Still there are few technical issues I would encourage the authors to address in order to improve and clarify the presentation.
These are given below in the section "Requested changes"

In summary, once the changes requested below are addressed the paper should be published.

The referee

1) Author stress the anisotropy of the g-tensor. I wonder of why this anisotropy is so significant, and what could be the origin of it.
2) i.e. attempt perhaps should read e.g. attempts
3) Is the full transparency of the interface a sufficient condition to set the two order parameters being equal in the superconductor and 2DEG? Could the author clarify when such assumption is a fair description of the proximity effect in the current setting?
4) On a technical side. Appendix B is meant to clarify how the condition of strong spin orbit compared to the gap and Zeeman splitting is used to invert the matrix needed to get the Green function. I am puzzled on which terms exactly are neglected in the Eq. (B2). The elements that are at first row and third column (1-3 element) connects the states differing in energy by 2 \lambda k and indeed can be neglected. Similarly 2-4 element can be ignored. Is that what is done or some other elements have been dropped as well? The element (1-2) connects states that might get degenerate for some momenta, so it seems one cannot neglect them. Am I missing something?
5) I am puzzled by the regime $g_{yx} \gg g_{xy}$. I thought that this tensor is symmetric. One can, for instance relate it to the second derivative of a suitable free energy with respect to the components of the field. Please clarify.
6) The departing model for the superconductor with quadratic field dependence of the Abrikosov-Gorkov parameter assumes the superconductor is thinner than the coherence length and not just than the penetration depth, as otherwise the order parameter acquires a noticeable spatial dependence. Is that the case for s system studied?
7) In the calculation of the DOS authors used the cutoff and the grid of 0.1\Delta in real frequency. Do one need a cutoff for DOS: the calculation is done per frequency. And is the grid fine enough to resolve the subtly features such as van Hove singularities?
8) Author show the reduction of the superfluid density at zero field by the disorder. Could the author compare it against a well-known results (the spin-orbit should not change it, I believe) ?
9) In the detailed fit to the data in Eq. (30) apart from the quadratic dependence of the depairing on the field the linear piece is needed as well. My question is could one interpret is as a signature of the Maki result for the bulk Hc2 in dirty limit?

In this manuscript the authors perform a detailed study of the emergence of a Bogoliubov Fermi surface in a proximitized 2D electron gas with Rashba spin-orbit coupling when it is placed in an external in-plane magnetic field. Most importantly, they include the effect of non-magnetic disorder solving self-consistently for the self-energy and show that the magnetic field magnitude range for which a gapless superconducting phase exists increases with larger disorder. Based on self-consistent solutions, they provide predictions for both the density of states as well as the superfluid density for different disorder strengths and magnetic field magnitudes. They also provide a detailed prescription for fitting the experimental data from recent work, enabling extraction of such parameters as anisotropic g-factors.

This work touches upon a very relevant and timely topic of Bogoliubov Fermi surfaces resulting from external magnetic fields or the supercurrent flow. The authors extended the current literature studies by self-consistent treatment of disorder and explained their calculation procedure in detail, which gives this manuscript a great pedagogical value. They also carefully explain the experimental fitting procedure, which I greatly appreciate. Therefore, the study certainly has a great value, and I could potentially see it published in SciPost Physics. One doubt I have is that according to the acceptance criteria, the work accepted in this journal is supposed to be a groundbreaking discovery or should open a new research direction, which this paper is not necessarily satisfying. Nevertheless, before making the final decision, I would like the authors to discuss the following issues:

1. The authors did not consider the parent superconductor that is the source of the proximity effect in their self-consistent calculation. While I agree that the inverse proximity effect on the parent superconductor may be of lesser importance, I am wondering if presence of such a superconductor would increase the robustness of the gapless phase beyond the very narrow range of magnetic fields in the clean systems. Would there be some sort of “pinning” effect that would stabilize the proximity-induced order parameter, especially if the magnetic field is smaller than what would close the gap in the parent superconductor? Doing the full calculation is probably too much work, but I would be grateful if the authors could provide some analysis of this issue.

2. Could the authors make some comparisons to the situation in which the topological Dirac surface state is proximitized, as was the case in the experiment in Science 374, 1381-1385 (2021)? This could be especially relevant since that reference claims to have observed Bogoliubov Fermi surface due to in-plane magnetic field. Do the authors expect that the impact of disorder is stronger in Rashba 2DEG as compared to the topological surface state?

3. Did the authors consider the possibility of the appearance of the in-plane vortices? In some superconductors like NbSe2 such vortices can appear at magnetic fields as low as 100 mT, smaller than the field that would lead to closing of the superconducting gap. How would the presence of such vortices impact the conclusions of the current analysis?

Minor points:
- Ref. 60 and 61 seem to refer to the same publication. Please remove the duplicate references.
- There are some issues with capitalization in reference titles, especially when considering chemical compounds, like in Ref. 62