InP/GaSb core-shell nanowires: a novel hole-based platform with strong spin-orbit coupling for full-shell hybrid devices

Dear Editor,

We thank you and the Referees very much for the review of our manuscript. From the Referees input, we have made some changes to the manuscript (see below) that, although do not change the results and conclusions of our work, clarify several issues and significantly improve the quality of our work. We believe these changes satisfactory address all the questions/remarks the Referees had, and thus it is suitable for publication.

Best regards,
Samuel D. Escribano on behalf of all authors

We refer to the responses to each referee to understand the motivation behind these changes. This is a comprehensive list with all the changes made:

- We have changed the title from "InP/GaSb core-shell nanowires: a practical proposal for Majorana modes in a full-shell hybrid geometry with hole bands" to "InP/GaSb core-shell nanowires: a novel hole-based platform with strong spin-orbit coupling for full-shell hybrid devices".

- We have modified/included two new sentences in the abstract. We no longer mention Caroli-de-Gennes subgap states, and we simply say "In addition, the SM wavefunction spreads all across the section of the nanowire, leading typically to a finite background of trivial subgap states with which MZMs may coexist". In addition, we include a better explanation of the SOC origin by saying "[...] that emerges from the combination of the intrinsic spin-orbit interaction of the SM active shell and the confinement effects of the nanostructure, thus depending...".

- In Section 1, at the end of the last paragraph, we have included the sentence: "We find that the SOC in these nanostructures originates from the combination of the intrinsic properties of the SM active layer and the radial confinement."

- In Section 2, in the fourth paragraph, we now include two more sentences. One is "However, different growing and fabrication methods, and particularly the ultimate incorporation of the SC outer shell in the hybrid NW, may change the position of this Fermi level. In this respect, we note that...". And the other one, at the end, is "Note also that other strategies to engineer the band alignment of hybrid quantum devices are currently being investigated. For example, in a recent experimental work [60], argon milling is used to modify the SC-SM interface while maintaining its high quality."

- In Section 2, at the end of the last paragraph, we clarify that "We make sure that our inhomogeneous FEM grid preserves the D6 symmetry of the hexagonal cross-section, which otherwise could introduce spurious solutions."

- In Section 3.2, in the last paragraph, we clarify that "Specifically, it is due to the radial confinement imposed by the nanostructure which breaks the translation symmetry of the crystal structure of this tetravalent SM".

- In Section 3.3, the third paragraph is new.

- In Section 3.4, the last paragraph has been extended.

- The title of Section 4 is now "Discussion and conclusions" and not only "Conclusions".

- In Section 4, in the first paragraph, we now clarify that "On the one hand, the tubular shape reduces the spread of the wave function across the NW section, confining it to the region close to the SC-SM interface. According to what happens in electron-based full-shell hybrid NWs, this should dramatically reduce the number of CdGM analogs coexisting with the MZMs [17]."

- In Section 4, in the second paragraph, we now explicitly say "Instead, it depends on the degree of HH and LH hybridization, ultimately regulated by the confinement strength provided by the NW radius."

- In Section 4, in the third paragraph, we include the sentence "Nonetheless, smaller radii require larger magnetic fields to achieve the topological phase, which may be detrimental to the parent superconductor. Therefore, a trade-off R must be chosen based on the SC material."

- In Section 4, the seventh and ninth paragraphs have been extended.

- In Section 4, we have included a new paragraph at the end.

- In the Appendix B, just before Eq. (39), we now clarify that "We note that this ansatz for the radial wavefunction is only valid for a core-shell NW with a thin w compared to R."

- Footnotes 5, 6 and 7 are new.

- Typos have been corrected, as well as some style issues.

- The references have been updated.