InP/GaSb core-shell nanowires: a practical proposal for Majorana modes in a full-shell hybrid geometry with hole bands

- alternative proposal for a full-shell nanowire structure with potentially larger spin-orbit and hence larer topological gap

- unclear description of the spin-orbit effect in the paper
- conclusions on the origin of the spin-orbit effect are doubtful

My apologies for the late report.

This paper deals with an alternative proposal for realizing a full-shell nanowire to create Majorana zero modes. The authors in particular argue that using holes in the valence band gives rise to a larger spin-orbit coupling and hence a larger topological gap.

The other referees have already mentioned several aspects related to using the core-shell nanowire in a full-shell structure for Majorana zero modes, and I agree with their assessment there. In addition, I find the discussion of the main result, the large spin-orbit coupling at times confusing and I am not convinced by some of the conclusions the authors draw in this paper. In particular:

1. Readability of the manuscript: The authors do not properly define what they mean with spin-orbit interaction. Since the system has inversion symmetry and time-reversal symmetry, there can be no k-dependent spin splitting of the bands, and indeed this also does not appear in the numerics. However, this is what a reader would expect when hearing the word Rashba SOI in isolation. The authors comment a bit on this on page 7 on saying the Rashba field is radial, but this is rather vague.
To make things worse, when the authors write down an effective Hamiltonian, e.g. Eqs. (1) and (9), the usual Rashba term $\alpha k_z \sigma_y$ arising from an electric field appears, which would lead to a k-dependent spin-splitting! However, this is *not* what e.g. Ref. [20] uses! There the Hamiltonian is $\alpha
\frac{\vec{r}}{r} \cdot (\vec{k} \times \vec{\sigma})$. This is deeply confusing. Is this just sloppy notation? Is the derivation of the effective model wrong?
2. Interpretation of origin of spin-orbit coupling: The authors write "These results lead us to conclude that the origin of the SOC is an inherent property of GaSb, rooted in the symmetry of the crystal structure of this tetravalent SM." I don't find this convincing at all. The model used by the authors is a generic k.p model that applies to a wide range of III-V semiconductors, i.e. their results do not seem specific to GaSb. Furthermore, for the effective model the authors find the effect even in spherical approximation, meaning that the effect does not depend on the crystal structure.
To me it seems more likely, that the band edge discontinuity at the core plays a similar role to the band bending in the previous studies (albeit now a discrete jump). The size of the effect can de bigger, as this for holes (similar to what is called direct Rashba SOI in Ref. [50]). To this end it would be interesting to see if the effect also appears as the size of the core shrinks to 0 (no core). This would mean to extend Fig. 3 to zero core radius (or w=R). Either there is a maximum of the spin-orbit and goes to zero as the core vanishes (then the origin is the discontinuity at the core) or there is a finite strength even for no core, and then this would mean the effect is is due to the confinement of the nanowire itself.

Since these questions affect the very core of the paper, I cannot formulate any recommendation to publish until they are resolved.

Ask for major revision

In this work, the authors theoretically study InP/GaSb core-shell hole-band nanowires as a candidate material for the semiconductor part of full-shell hybrid superconductor-semiconductor (SC-SM) Majorana nanowires. The band structure of the bare InP/GaSb core-shell nanowire close to the topmost valence band is obtained from an 8-band Kane model through a self-consistent solution of the Poisson equation. The main finding is that the GaSb holes exhibit a strong intrinsic spin-orbit interaction that does not rely on electric fields. The authors also calculate the size of the ‘topological chemical-potential window’, which is the energy splitting between the two highest-energy hole subbands with total angular momentum quantum number |mF|=1/2. Based on their results, the authors argue that InP/GaSb core-shell hole-band nanowires are a promising candidate platform for full-shell Majorana physics.

The paper is interesting and well-written, and the presented band structure calculations for the bare InP/GaSb core-shell nanowire appear to be technically sound. However, I have a couple of concerns/questions related to the Majorana part of the work that should be addressed before I can recommend publication in SciPost Physics:

1) I think that the current title—and to some extent also the abstract—of the manuscript do not reflect its content very accurately. The reasons for this are the following:

- The paper does not discuss the actual Majorana physics in the SC-SM full-shell hybrid nanowire in any detail, but only the normal-state band structure of the bare InP/GaSb core-shell nanowire, which is why I think it is not appropriate to call the paper a “proposal for Majorana modes”. In particular, the authors neither write down a superconducting pairing term nor a time-reversal symmetry breaking term (magnetic flux), both of which are necessary for the emergence of Majorana zero modes in full-shell SC-SM hybrid nanowires. Instead, when it comes to the emergence of Majorana modes, the authors just cite a couple of previous works without any additional explanations. If the paper should be called a “proposal for Majorana modes”, I think it would be important to (i) explicitly specify the full Hamiltonian of the SC-SM hybrid, and (ii) substantiate the Majorana part of the work by additional calculations. Alternatively, the title of the paper could be changed.

- I also think the use of the word ‘practical’ in the title is slightly misleading. It appears that InP/GaSb core-shell nanowires have not yet been fabricated, and even if they can be fabricated in the future, it is not clear whether their quality will be high enough to overcome the usual disorder problems that plague more conventional (e.g. InAs-based) Majorana nanowire platforms. It is also not clear which superconductor can be used to induce a (hard) superconducting gap for the holes in the GaSb shell, nor is it known what the size of the induced gap would be. Taking all of these points together, the setup proposed here should in my opinion not be called ‘practical’ as none of the necessary ingredients are currently available in the lab.

- The abstract gave me the impression that CdGM states (and how their presence can potentially be reduced in a core-shell geometry) would be discussed in this paper, but this is not the case. Instead, CdGM states only enter the paper through references to previous works (mainly Ref. [17]). Maybe the authors can slightly rephrase the abstract in order to avoid this potential misunderstanding.

2) One important assumption made by the authors is that “the Fermi level is placed close to the VB edge of GaSb as reported in Ref. [28].” However, is there any reason to assume that the Fermi level remains close to the valence band edge in the presence of a superconducting shell? It does not seem unlikely that the Fermi level is shifted substantially in the presence of an SC shell (especially in the limit of strong SC-SM coupling, which is often the experimentally relevant limit). If this is the case, it is not clear whether the chemical potential can ever fall into the ‘topological chemical-potential window’ calculated by the authors (see Fig. 4 in the manuscript) since the Fermi level cannot be adjusted by gating in a full-shell geometry. It would be good if the authors could comment on this potential problem.

3) The radii of the wires considered in this work (R=15-40 nm) are relatively small compared to the radii considered in previous works on InAs electron nanowires (e.g. R~65 nm in Ref. [9]). For such small nanowire radii, the magnetic field needed to thread one flux quantum through the SC shell can become sizable. Can the authors give an estimate for the approximate field strength that is needed to tune their system deep into the topological phase? I wonder how much the band structure shown in Fig. 2 will be modified by the magnetic field in the SM if both Zeeman and orbital effects are taken into account. Especially orbital effects might be important in a core-shell geometry, see also point (4) of the first Referee.

4) I think there may be a typo on page 7: "As in the case of Ref. [20], we find that α increases with R." Shouldn't it be "[...] decreases with R"?

Ask for minor revision

Vezzosi et al. investigate theoretically using hole bands in core-shell InP/GaSb nanowires incapsulated in a full superconducting shell. Full shell nanowire setups have the advantage of not requiring strong Zeeman energies, rather relying on orbital effects to induce a topological phase. Nonetheless, such setups suffer from the usual issue of distinguishing topological states (MZMs) from trivial. In particular, in the case of full shell nanowires, Caroli–de Gennes–Matricon states can give zero-bias peaks (ZBPs) reminiscent of MZMs. In their paper Vezzosi et al. propose that hole bands in InP/GaAs full shell nanowires could be a better platform than the previously used InAs setups [see Vaitekėnas et al. Science 367, 6485 (2020)], which have attracted both skepticism and controversy. The proposed setup using holes in this paper has the advantage of strong spin-orbit coupling (SOC) than InAs and does not rely on strain or electric fields.

The calculation in this work is solid and a nice example of how core shell nanowires and holes could be a good strategy for achieving strong SOC and, potentially, topological superconductivity. I am minded to eventually accept for SciPost, however I have some concerns since the authors are proposing a very specific platform. In particular, a more in depth discussion of the SC is required before it is possible to really assess the validity of this proposal [see (1) below)]. Nonetheless, if the authors can provide a convincing discussion of the SC then I believe this paper should be published in SciPost and hopefully will encourage experimental efforts.



(1) The authors state “in this work we make a specific and practical proposal“, however throughout — until the final paragraph — I had the question in my of which superconductor was part of this proposal (SC of course being vital for achieving TSC). It is stated at the very end “In any case, the main conclusions of our work should not be affected by the particular SC”. I am not sure I completely agree. Could the authors comment on the following:

1.1) It is stated that metallization effects could be especially important in this setup. I agree with this and think it would be worth spelling out that metallization effects will e.g. likely effectively increase the effective R of the wavefunction and therefore reduce SOC. Similar effects have been seen in before semiconductor and topological-insulator nanowires. I do not think that metallization will negate the benefits of the proposed setup in this paper, but a more in depth discussion of its impact would be useful.
1.2) The TSC region is more than an order of magnitude larger than achieved in the standard Lutchyn-Oreg model, which is the main merit of this proposal. However, Fig. 4 gives the impression that the SC does not matter so much. It would be useful here to briefly mention why it is still beneficial to have a large SC gap (e.g. well localised MZMs etc).
1.3) Although less familiar with attempts to induce superconductivity in GaAs, I worry that many SCs (e.g. Al) will result in a considerable increase in disorder of the GaAs due to the usual mechanisms: diffusion, cross hatched patterns, and spatially dependent interfaces. In assessing the viability of this proposal, it would be very useful for the authors to discuss past literature where such proximity effects have been attempted.

(2) A further concern with the combination of metallization and full shell nanowires is the lack of control over the location of the chemical potential. Other than the larger TSC region (a few meV), it is not really clear to me how this proposal mitigates that issue. Could the authors comment on whether this platform has any benefit (or is further hindered) by the lack of control over chemical potential compared to, say, InAs full shell nanowires?



(3) Another worry is that the limited gating possible in the proposed devices will mainly affect the ends of the nanowire and so will alter the wavefunction primarily in the region at the end. In the proposed setup the conjunction of gating with the likely dependence of metallization on the exact spatial extent wavefunction, seems likely that it could result in many parameters that are rather smooth at the end of the nanowire, which would result in quasi-MBSs. Could the authors comment on if they agree and how to mitigate this? Or do they intend the experiments to be done without any gating?

(4) The extension of the wavefunction around the nanowire core is also reminiscent of topological insulator nanowire. There it was shown that there can be issues inducing superconductivity close to a half-flux quantum, depending on the presence or absence of a vortex in the superconductor and the orbital effect of the states in the TI nanowire [de Juan et al. SciPost Phys. 6, 060 (2019)]. Here it seems there will also be an orbital effect not only in the full SC shell (causing a vortex), but also in the semiconductor. The period of the orbital effects in SC/SM will be different but both related to the radius of the nanowire. I worry that the induced superconductivity might be harmed by the orbital effect of the state in the core-shell nanowire. Could the authors comment on the impact of the orbital effect on induced SC in the nanowire?

(5) Finally, could the authors comment on whether the main experimental signatures of MZMs in the devices they envisage remain those of previous studies: Namely, ZBPs and Coulomb blockade spacing or if there are new signatures possible due to the core-shell nature of the nanowire?

Ask for minor revision