Conventional s-wave superconductivity and hidden peak effect in single crystals of Mo$_8$Ga$_41$ superconductor

Mo8Ga41 is a potentially unconventional superconductor that has been controversially discussed in the previous literature. The manuscript under consideration reports the very first penetration depth measurement that demonstrates an unexpected but remarkably good agreement with the BCS expression in the clean limit. This interesting result sheds new light on the physics of Mo8Ga41 and potentially meets the SciPost Physics criterion of presenting a breakthrough in an existing research direction. Moreover, a curious peak effect in the critical current is reported and analyzed. I believe that this manuscript could be published in SciPost Physics, but it requires a major revision along the following lines:

1. One major puzzle is the simple BCS behavior of the superfluid density, as opposed to several previous observations of the "less conventional" superconductivity in Mo8Ga41. Some of these reports could be affected by the ambiguous fitting of the muon data or by surface effects, but bulk probes such as heat capacity should be more tenable. The large jump in the specific heat at Tc [PRB'2016] does indicate some physics beyond the weak-coupling limit of BCS. This observation directly contradicts the conventional scenario advocated by the present work. A plausible explanation for this discrepancy should be given.

2. One plausible explanation is the sample dependence. In this regard, details of the sample characterization are essentially missing, and it is not even clear whether exactly the same crystal(s) as in Ref. [14] have been used. I strongly encourage the authors to provide additional characterization data, such as:
- resistivity
- magnetization
- heat capacity
that could be compared to the previous publications. It would also be natural to supply Fig. 8 with the Bc2 values from the existing literature. Do they match?

3. Another possibility is that the superconductivity of Mo8Ga41 is not of the simple BCS type, but a more complex scenario mimics the BCS temperature dependence of the superfluid density, such that the good match in Fig. 2 is merely accidental. This idea may not be so odd if the physics of the normal state is considered. Mo8Ga41 is far from being a simple metal. Its resistivity exceeds the MIR limit and shows a rather peculiar temperature dependence below 100 K. The mean-free path should be quite short. Would one really expect the clean limit to be applicable in this case? Should not the dirty limit be more appropriate? What is the mean-free path in your sample?

4. The lower critical field was determined by the NV-center magnetometry. The Bc1 value at 4.25 K is somewhat lower than reported in PRB'2016: 85 Oe vs. 110-115 Oe. What is the reason for this difference? How was the sample surface prepared?

5. On page 5 there is a typo in the description of the critical current at 2 T and 4 T in the FC and ZFC regimes. The sentence with "the ZFC curves reverse this order" should probably read as "the FC curves reverse this order". According to Fig. 6, the order is reversed below 4 K only. Why does it not happen at higher temperatures? Would it be possible to shed more light on the vortex state of Mo8Ga41? For example, is there a vortex liquid, and in which temperature range?

Ask for major revision

The manuscript comprises a comprehensive study of magnetic field penetration to a single-crystalline Mo_8Ga_41 superconductor. It brings well presented, original and interesting results .

In the manuscript sometimes shortcuts are present, some parts are confusing.

I would recommend the publication after the authors clarify/correct the issues listed below.

A) Technical remarks, I would recomend to:

A1 - use proper expression for the ratio of the energy gap and the critical temperature (involve Boltzmann constant), especially when you refer to specific numbers.

A2 - try to avoid mixing SI and CGS units in the same text. CGS is obsolete - it is easy to use mT instead of Oe.

A3 - in some parts of the text there are references to numbers of the sections (Section II, Section III.B) even if the sections are not numbered. It seems that it is some residuum from an older manuscript layout. Specially refering to Section III.B in a third section and second subsection is confusing.

B) The introductory part seems to be confused:

B1 - The authors claim that in Ref.9 the larger gap gives the ratio Delta/k_BT_c =1.857. However, this was calculated considering the smallest value from the distribution of the larger gaps (1.45 meV), moreover, local T_c of the junction with this energy gap was not determined. On the contrary, the temperature dependence of the two energy gaps with clearly identified critical temperature presented in Ref.9 gives the value of Delta 1.6, that leads to the ratio of 2.06, which is larger than the weak-coupling limit.

B2 - The authors claim that in Ref.16 the larger from the two gaps has the ratio even smaller than the weak-coupling limit. This is not true. Actually it is around 5, which is really unexpectedly far above the weak-coupling limit.

B3 - The authors comment on the results presented in Ref.17, stating that the data can be fit well both with single-gap and two-gap fit with minor differences. This is true, but the main message of the paper is that none of the options is valid, because the single-gap fit leads to the ratio much lower than the weak-coupling limit and the two-gap fit is not consistent with the heat capacity measurement performed on the same crystal. Instead, the paper is a warning that the measurement of the superfluid density itself, without knowing the complex picture, might be misinterpreted e.g. for a two-gap feature. This context is not mentioned in the Introduction.

C) In the section Samples and methods, subsection Lower critical field, the authors claim that the measurements are performed as close to the sample edge as possible and refer also to Ref.17. However, in Ref.17 it is explained that for the measurement of the penetration field in this sample featuring strong pinning (V-shaped profile), a probe is selected that is distant from the edge by the half-width of the sample. This does not mean "as close as possible", actually the selected probe was the forth in a row from the edge (see Fig.1 in Ref.17). I would suggest to elaborate this part - mentioning the sample shape and sizes, and estimate the position of the measurement in respect to the sample edge. Was some other position examined as well?

D) I wonder why the authors used weak-coupling limit for calculation of lambda. The other literature refers mostly to moderate or strong coupling (Ref.9: Delta/k_BT_c =2.06; Ref.15: jump in Cp 2.83>1.43; Ref.16: Delta/k_BT_c =2.1 if single-gap fit is considered; Ref.17/18: Delta/k_BT_c =2.2). What would be the difference in the penetration depth and subsequently in the superfluid density (since one needs to feed the value of lambda(0) for the calculation), if larger coupling was involved?

E) In the section Results and Discussion, subsection London penetration depth..., there is a sentence I consider to be too simplistic: "There is no indication of a multi-gap behavior, which usually appears as a convex curvature ..." The shape of the superfluid density depends on various parameters - mostly on how far are the two energy gaps apart and what is the weight of their contribution. It is true, that most of the clearly identified two-gap superconductors have this feature, but it is not general. In other words, absence of the positive curvature does not necessarily exclude two-gap superconductivity.

F) Decription of Fig. 3 looks confusing - the Y-axis is called lambda_m, in the figure caption it says that it shows Lambda_m(T)*lambda(0)+DeltaLambda(T) and in the text it states "note that the measured DeltaLambda_m(T)..." Also the notion of saturation is rather confusing, since all the curves in Fig.3 increase at elevated temperatures, while their values overlap.

G) The first sentence of the section Results..., subsection Campbell penetration depth... looks confused, probably with wrong position of the bracket.

Ask for minor revision