Conventional single-gap $s-$wave superconductivity and hidden peak effect in single crystals of Mo$_{8}$Ga$_{41}$ superconductor

Dear Editors, We appreciate the thoughtful and detailed comments from the referees. Here, we provide point-by-point answers and indicate the changes in the manuscript. We follow the numbering of the REPORTS.

REPORT 1

1) In response to comment B3, …..

Yes, there is a significant disagreement between the superfluid density and the specific heat, both in Verchenko 2017 and in Marcin 2021. We note that the superfluid density is a quantity directly connected with the superconducting gap structure, whereas total specific heat has multiple contributions from all possible excitations in the complex system. The electronic part is notoriously difficult to separate. Typically used quenching superconductivity by a magnetic field is problematic due to magnetoresistance in this system [new reference: Zhao 2020 in the revised manuscript]. The unusual specific heat is an interesting problem, but it is beyond the scope of the present work. Furthermore, the Referee’s own excellent literature analysis in section #3 lists several issues, both technical and fundamental. We agree with that statement.

We followed the advice of the Referee and reorganized and rewritten the introduction to discuss these issues, highlighting the work of Verchenko et al. and Marcin et al. In light of this discussion, we believe that our work contributes significantly to the unusual physics of the Mo8Ga41 superconductor.

2) On top of that, I think ...

One has to focus on the superfluid density obtained using London penetration depth, which is shown in Fig.111 of Carbotte’s paper, Ref. [24] of the revised manuscript. The Referee is correct that for a gap to Tc ratio of 2.2, the strong-coupling parameter Tc/omega_ln would be around 0.1, and the resulting superfluid density will be close to the dotted curve for 0.15 in Ref.[24]. In our view, the difference between this curve and the weak-coupling BCS is significant and would easily be observed in our measurements. Furthermore, the suggested two-band scenario has to be solved self-consistently within Eliashberg theory and not using an unphysical alpha-model. We address this in the updated introduction.

3) I think it is important...

We agree with this analysis, which nicely connects several works. We have thoroughly rewritten this part of the Introduction, focusing on the superfluid density vs. specific heat reported in the same papers, and discussed the shortcomings of the models used for the analysis. We have also addressed the important question of clean vs. dirty limit.

4) Comment D in the previous report …

In the clean limit, \lambda(0) is a normal state property and does not depend on any superconducting parameters. In the presence of scattering, it is renormalized but only weakly. According to Tinkham, \lambda(0) = \lambda(0)_clean*\sqrt(1+\xi_0/\ell) where xi_0 is the BCS coherence length and \ell is the mean free path, so the correction is relatively weak. It is more important, though, that \lambda(0) in our paper was obtained from direct and independent measurements using NV magnetometry in a model-independent manner. Therefore, our superfluid density is obtained from the experiment without any assumptions, and it is fully consistent with the weak-coupling BCS.

5) There is still a Boltzmann constant ...

We are sorry that we missed it in two places. It is corrected.

REPORT 2

1. Please, write explicitly ...

In the original text we describe how the crystals used in our study were obtained, summarizing the protocol and giving the reference: “\textbf{Samples:} Single crystals of Mo$_{8}$Ga$_{41}$ were grown by the high-temperature self-flux method. Mo and Ga were mixed in an 8: 500 ratio in an alumina crucible and sealed in an evacuated quartz tube. The ampule was heated up to 850$^{0}$C in two hours, held at 850$^{0}$C for 10 hours, and then slowly cooled to 170$^{0}$C for 55 hours, when the crystals were decanted \cite{Petrovic_QO_2020}.” For added specificity and to address the Referee’s concern, we now added at the end of this paragraph: “The full details of crystal growth and characterization are given in Ref.\cite{Petrovic_QO_2020}.”

1. Please, provide additional evidence ...

As the Referee pointed out, our superfluid density is very similar to the muSR data, since both can be fitted well by the weak-coupling BCS formula. Regarding the Hc2, we followed the Referee’s advice and now included the data not only from Verchenko 2016, but also from Marcin 2019. Both agree with our Hc2(T) data. Figure 8 is modified, and the text includes proper references.

1. I agree that the observation of ...

We found additional information about the residual resistivity ratio and now write regarding the clean limit: “…Mo$_{8}$Ga$_{41}$ appears to be rather clean, as is evident from the residual resistivity ratio of 15.4 \cite{Marcin2019} and, importantly, from the observation of quantum oscillations \cite{Petrovic_QO_2020}. (Our data are also consistent with the clean limit).”

and in the discussion:

“As mentioned in the Introduction, the clean limit is independently supported by observations of quantum oscillations in Mo$_{8}$Ga$_{41}$ crystals \cite{Petrovic_QO_2020} and a not too low residual resistivity ratio of 15.4 \cite{Marcin2019}.” We thank the Referee for pointing out this interesting reference. We now cite it in the paper in the introduction of the revised text: “A quantum oscillations study of Mo$_{8}$Ga$_{41}$ inferred three-dimensional electronic bands with strong coupling to phonons \cite{Petrovic_QO_2020}. Similarly, a $T-$linear resistivity observed in a wide temperature interval was interpreted to be due to scattering from low-lying phonon modes, which could also lead to enhanced electron-phonon coupling \cite{Zhang2020}.

REPORT: I would also like to mention that the other reviewer's comments ...

Please, see our reply to REPORT 1, #3 above. We reorganized and rewritten the introduction by including the suggested discussion.

Major changes:

The introduction was thoroughly revised following the points raised in the REPORTS.
Figure 8 was updated to include a comparison with literature Hc2.
The sample synthesis part is updated.
Introduced eleven new references.