Non-Topological Edge-Localized Yu-Shiba-Rusinov States in CrBr$_3$/NbSe$_2$ Heterostructures

For the record, I am posting here the responses of the authors to the four referee reports. Carlo Beenakker, editor in charge.

Referee 1 (report 2)

1. Given the importance of the atomic and electronic structure of the island edges, Fig. 3b would benefit from a larger presentation. Reply: We agree with the referee that Fig. 3b, in the original manuscript, was quite small. In the revised version, it has been enlarged. To achieve this, the panels of Fig. 3 have been slightly reordered.

2. Is there any spectroscopic indication of the enhanced DoS of the Cr 3d states at the edges? Reply: The observation of an enhanced DoS at the edges would strengthen our suggestion of the origin of the edge-localized YSR states. Unfortunately, instabilities of the tip-sample junction complicated the spectroscopic characterization of the edges at higher biases. As such, we cannot presently make definitive statements about the Cr 3d DoS at the edges.

Referee 2 (report 3)

1. It can be nice adding a short discussion about the potential role of strain or different twist angle for the absence of bulk Yu-Shiba-Rusinov states. Reply: We agree with the referee that the role of strain and twist angle should be investigated in more detail. On page 5, line nr. 131, our original manuscript hinted at this via the sentence: “The moiré pattern is not exactly aligned with the rows of Br atoms, showing that the twist angle is not exactly 30° (as suggested in previous reports), but more close to 32°.”

We have modified the final paragraph of the Conclusion to include a brief discussion on this: “The difference may be caused by a different substrate-epilayer interaction (e.g., induced by strain) or, possibly, the presence of defects/impurities. Furthermore, here we found a twist angle of 32°, whereas an angle of 30° was reported previously. A different twist angle may affect the coupling between the ferromagnet and the superconductor.”

Referee 3 (report 1) Did not recommend/ask for any modifications.

Referee 4 (report 4)

1. My main criticism is of the YSR treatment in this paper. It was confusing for me if the authors claim this is a quantum spin or classical spin system. They in the end use a spin ½ treatment to explain their data (e.g. Fig. 5), which is not appropriate in the case that there is a high spin. The data could also be understood in terms of having different types of isolated YSR states of a high-spin character. Experimentally they observe multiple peaks, and they cannot exclude the possibility that this is a high spin YSR state (as opposed to some band). In both cases, there is extensive theoretical and experimental literature, and I do not think they can claim to understand this in the simplest limit. For example, changing the exchange field, e.g. in Fig. 5b, may also be highly non-linear in the limit of high spin (see e.g. PRB, 105, 235406 (2022), and PRB, 103, 205424, (2021)). I can understand that the spin ½ picture is the easiest to interpret, but I believe that it is important considering they observe something in contrast to published reports, to also loosen this claim and explain the other theoretical interpretations of the data (e.g. claiming a QPT).

Answer: It is not our objective to claim that the observed YSR states are best described using quantum or classical models. Instead, our focus is the absence of (1) a robust zero-energy edge state, and of (2) YSR bands in the island interior (which both have been observed in previous work). We merely employed the classical spin model to present possible explanations of our findings, including the variation of the YSR state energy as a function of tip-sample distance. However, we agree with the referee that other models, e.g. based on quantum spins, may explain our findings as well. We therefore added the following sentence to the discussion of Fig. 5c (Page 8, line 244).

“We cannot exclude that, as the tip-sample distance becomes smaller, the charge transfer between NbSe2 and CrBr3, as well as the magnetic moments may change, which could also contribute to the experimentally observed behavior. This would be especially relevant for quantum spin systems with magnetic moments above S=1/2.”

1. I missed a bit of a more in-depth discussion at the end of the paper to describe the differences between this work and the previous papers.

Answer: This point was also raised by referee 2 (report 3). In the first revised version, the Conclusion paragraph was extended with a brief discussion outlining the potential importance of twist angle, strain and defect/impurity effects.

1. I found a bit confusing: is the CrBr3 layer insulating? Do they have experimental evidence of this? For example: wouldn’t this suppress Andreev reflections (e.g. in their shot noise discussion)?

Answer: Yes, CrBr3 is a ferromagnetic insulator. Note that the STS spectrum acquired above CrBr3 does not show additional features compared to that measured above NbSe2 (see Figure 2a). Indeed, in planar junction transport experiments, the presence of an insulating layer may suppress the occurrence of Andreev reflections, as it increases the tunnel barrier strength and decreases the junction transparency. However, we perform our shot-noise measurements at constant junction resistance (page 3, line 94). Hence, the presence of an insulating CrBr3 layer does not the suppress Andreev reflections in our shot-noise experiments.

1. Could the authors specify their experimental resolution and the tip type?

Answer: A mechanically polished PtIr tip was used in the experiments, as is written in the methods section of the manuscript (page 3, line 75). The electronic temperature in our system, as estimated from the width of the coherence peaks is approximately 500 mK. We have added a sentence to the methods section (page 3, line 74).

1. Is hydrogen an important consideration, for example in comparing differences?

Answer: We do not think that the magnetic properties of our CrBr3/NbSe2 are affected by hydrogen. Hydrogen-dosing experiments have never been performed in our setup, and care is taken not to run titanium sublimation pumps in vicinity of sample preparations. Additionally, we did not observe any mobile species on the surface, which could be indicative of molecular H2.

1. In Fig. 1, what are the clusters on the edge of the island? Won’t this create disorder?

Answer: We cannot make any definitive claims about the nature of the bright spots observed in the topography images of the island edges. The apparent height of these features is only slightly higher than that of the interior of the islands (less than 50 pm difference, see Fig. 1c). Whatever the origin, there is indeed disorder. However, we note that topological superconductivity should not be affected by this type of disorder. The fact that we see in-gap states with spatially varying energies (for which disorder may partially be responsible), supports the assignment of the in-gap states to topological trivial states (in contrast to previous work).

1. Did the authors apply magnetic field to quench the superconducting state and see if there’s residual LDOS at the island edge, i.e. if it’s metallic? I found this confusing, as e.g. p7 the authors write that the “increased DOS strengthens the exchange interaction…” If it’s insulating, it isn’t clear why this would matter.

Answer: The phrase mentioned by the referee refers to the outcome of DFT simulations on the related heterostructure CrCl3/NbSe2, which is part of a paper that is currently under review. From those DFT simulations, we observed a slight increase in the LDOS at the island edge that coincides with an increase in the exchange interaction strength. To make this more clear, we have added a sentence on page 7 (line 182):

“For that material system, we find that at the edges of the island the LDOS at energies close to the Fermi level is increased, strengthening the exchange interaction between the Cr atoms and the NbSe2 substrate.”

1. Could the authors discuss other things that may change if the layer relaxes when the tip is approached? Hypothetically, there can be a different charge transfer, magnetic moment, or ultimately magnetic anisotropy: all matter if it is beyond spin 1/2.

Answer: We agree that the factors mentioned by the referee would be relevant if the material is best described by a quantum spin model. We used relatively low setpoint currents in many of the reported experiments. In these cases, the tip is relatively far away from the heterostructure, and the tip is not expected to lead to (significant) changes in the CrBr3/NbSe2 interaction. For completeness, we added the following sentence (page 8, line 244) to point out that other factors may also be relevant. Finally, we would like to point out that these factors do not change the main conclusions of the manuscript.

“We cannot exclude that, as the tip-sample distance becomes smaller, the charge transfer between NbSe2 and CrBr3, as well as the magnetic moments may change, which could also contribute to the experimentally observed behavior. This would be especially relevant for quantum spin systems with magnetic moments above S=1/2.“

Other changes: Since the submission of the manuscript, we have revised the method to extract uncertainties in the effective charge from shot-noise measurements. This revised method results in smaller uncertainties. We reanalyzed the shot noise data using the updated method (calculating the uncertainties from the uncertainty in the measured power spectral density directly) and updated Figure 6b and 6c, and the accompanying discussion. The outcome remains unchanged: the updated value of Qeff inside the gap is 1.67 +/- 0.02 e (was 1.7 +/- 0.1 e).