Strong-Coupling Behavior of the Critical Temperature of Pb/Ag, Pb/Cu and Pb/Al Nanocomposites Explained by Proximity Eliashberg Theory

Round 1

Reviewer 1 Report

This paper investigates the effect of proximity coupling between superconducting bulk lead and a non-superconducting metal (either copper, silver or alluminum). The main motivation for this work traces back to PRB 76 144510, where the Authors (disturbingly) state that Pb (a strong coupling superconductor) behaves like a weak coupling superconductor when proximity linked with silver. G. Ummarino, here, presents some simulations using Eliashberg theory (strong coupling) to show that an excellent agreement with the available experimental data can be achieved without any unphysical assumption about bulk lead. I find this work to be valid and correct.  

  

However, from the computational point of view, this work is a (almost) trivial application of a classical approach by Stephan and Carbotte of Eliashberg theory in proximity coupled systems. Per se, the results are not worth of a publication. The added value should be the fact that the Author clarifies how a (say) correct approach should be used to describe proximity effects.  For this to be effective the introduction and discussion should be extended in the form of a brief but clear review. If this is done I believe that the paper should be accepted for publication. 

There are a few points that I would like to see discussed:  

1] It is stated that the coherence lenght is the characteristic lenght scale of the proximity effect [A similar statement can also be found in ref 5]. However the coherence length is defined via the electromagnetic response of the superconductors, which is conceptually a very different property [see for example the Rev.Mod.Phys of Carbotte]. I think that this aspect should be discussed in detail, i.e.: 

-Is there a rigorous way (in Eliashberg theory) to prove that the coherence length is also linked to proximity? 

-If not in Eliashberg theory, is there at least a BCS or a model explanation why the two length scales should be equivalent. 

2] How is the proximity theory connected with the (way more popular) theory of multi-band superconductivity? Conceptually the two approaches  appear to be quite similar (apart from the origin of the pairing). It would be nice if the differences were discussed here.

3] It is mentioned (page 3) that the result doesn't depend on the product A|t^2|. But clearly at A=0 there would be no coupling at all. And Tc would be that of bulk Lead. There must be a regime where this statement breaks down. In general I suggest to insert a discussion on the limits of applicability of the theoretical framework. 

 4] In the above mentioned Ref. 5, proximity effect is discussed in terms of Andreev reflection. I'm not sure if this is correct. In case the Author should discuss how Andreev reflection enters in the formalism. 

 

Author Response

My reply is in red

This paper investigates the effect of proximity coupling between superconducting bulk lead and a non-superconducting metal (either copper, silver or alluminum). The main motivation for this work traces back to PRB 76 144510, where the Authors (disturbingly) state that Pb (a strong coupling superconductor) behaves like a weak coupling superconductor when proximity linked with silver. G. Ummarino, here, presents some simulations using Eliashberg theory (strong coupling) to show that an excellent agreement with the available experimental data can be achieved without any unphysical assumption about bulk lead. I find this work to be valid and correct.

I thank the referee for his observations to which I will try to reply.

However, from the computational point of view, this work is a (almost) trivial application of a classical approach by Stephan and Carbotte of Eliashberg theory in proximity coupled systems. Per se, the results are not worth of a publication. The added value should be the fact that the Author clarifies how a (say) correct approach should be used to describe proximity effects.  
For this to be effective the introduction and discussion should be extended in the form of a brief but clear review. 
If this is done I believe that the paper should be accepted for publication. 

There are a few points that I would like to see discussed:  

1] It is stated that the coherence lenght is the characteristic lenght scale of the proximity effect [A similar statement can also be found in ref 5]. 
However the coherence length is defined via the electromagnetic response of the superconductors, which is conceptually a very different property [see for example the Rev.Mod.Phys of Carbotte]. I think that this aspect should be discussed in detail, i.e.: 

Perhaps more than the coherence length for the electromagnetic response we should refer to the penetration length.

-Is there a rigorous way (in Eliashberg theory) to prove that the coherence length is also linked to proximity? 
-If not in Eliashberg theory, is there at least a BCS or a model explanation why the two length scales should be equivalent. 

The proximity effect theory was developed by Mac Millan (W.L. McMillan, Phys. Rev. 175, 537, (1968)) in the BCS framework and then generalized by  Schachinger and Carbotte to Eliashberg's theory 
(E. Schachinger and J.P. Carbotte, J. Low Temp. Phys. 54, 129 (1984)). In MacMillan's theory the characteristic length of the proximity effect is the coherence length that is the typical size of the Cooper pairs 
and the proximity effect is connected with the Cooper pairs inside the normal metal.

2] How is the proximity theory connected with the (way more popular) theory of multi-band superconductivity? Conceptually the two approaches  appear to be quite similar (apart from the origin of the pairing). It would be nice if the differences were discussed here.

A profound analogy exists between the proximity system and the two-gap model. If we assume that in the second band, as in the normal film, there is no intrinsic pairing (for example as it happen in the magnesium diboride)
we have induced superconductivity i.e.an induced energy gap appears. The substantial difference in these two situations is
that the two-band model the bands are separated in momentum space and the second band acquires an order parameter due to phonon
exchange while in the proximity effect the systems are spatially separated, and superconductivity is induced by
the tunneling of Cooper pairs. In the first case the coupling is in the k-space while in the second is in the real space but the mathematical formalism is the same.
Finally also the effect of a static electric field on the critical temperature of a superconductor
can be explained in the framework of proximity Eliashberg theory (GA Ummarino et al, PRB 96 064509 (2017))

3] It is mentioned (page 3) that the result doesn't depend on the product A|t^2|. But clearly at A=0 there would be no coupling at all. And Tc would be that of bulk Lead. There must be a regime where this statement breaks down. In general I suggest to insert a discussion on the limits of applicability of the theoretical framework. 

Yes, we agree with the referee: the product |t|^2 A has to be independent from the area. 
However it is possible to write Γ also in another way where A does not appear, as it is possible see in W. McMillan Phys Rev 175, 537 (1968). 
In any case we have verified that in a very wide range of values of A the final result is independent of the aforementioned value. 
Only in extreme cases numerical problems could arise which do not allow obtaining a solution of Eliashberg's equations

4] In the above mentioned Ref. 5, proximity effect is discussed in terms of Andreev reflection. I'm not sure if this is correct. In case the Author should discuss how Andreev reflection enters in the formalism.

The role of Andreev reflection (T.M. Klapwijk, Journal of Superconductivity: Incorporating Novel Magnetism, 17, 593 (2004)) is fundamentals in the undestanting
of microscopic mechanism at the origin of proximity effect. It happen that single electron states from normal metal are converted to Cooper pairs in the superconductor. The proximity effect can be seen as the result of interplay between long range order inside the normal metal and Andreev reflection at the normal metal-superconductor interface (B Pannetier and H. Courtois, Journal of Low Temperature Physics, 118, 599 (2000)).
The link between Andreev reflection and proximity effect exist because the Andreev reflection of an electron or a hole is equivalent to the transfer of a Cooper pairs in or out of the superconductor i.e. to presence of Cooper pairs inside the normal metal.

All these considerations have been included in the paper.

Reviewer 2 Report

This paper computes Tc of composites of Pb with Ag,Cu and Al using Eliashberg theory. For the non superconductors, Ag and Cu an ad hoc assumption is done where the dos of the metal is replaced by that of the superconductor. The motivation is purely the agreement with the experiment. For Al this replacement is not needed. The paper can be published after the author provides some plausibility argument for the above assumption. Also for completeness in the case of Al the curve with the dos replacement should be given.

Author Response

My reply is in red

This paper computes Tc of composites of Pb with Ag,Cu and Al using Eliashberg theory. For the non superconductors, Ag and Cu an ad hoc assumption is done where the dos of the metal is replaced by that of the superconductor. The motivation is purely the agreement with the experiment. For Al this replacement is not needed. 

I thank the referee for his observations to which I will try to reply.

The paper can be published after the author provides some plausibility argument for the above assumption. 

If the dimensions of the nanoparticles are only smaller than the coherence length of the superconductor (i.e. the dimensions of the Cooper pairs) 
it is possible that the electronic properties of the nanoparticles are replaced by those of the superconductor.

Also for completeness in the case of Al the curve with the dos replacement should be given.

The curve with the dos replacement has be calculated.
Here it is very clear that this method doesn't work. The reason is simple: in this case the standard theory works because the Al nanoparticles sizes are greater and therefore the starting assumption is no longer valid.

All these considerations have been included in the paper.

Reviewer 3 Report

This work shows that the experimental critical temperature of nanoparticle systems, including superconducting Pb and normal Ag, Cu, and Al materials, is governed by the proximity effect and can be accurately explained using Eliashberg's theory. It refutes the initial weak coupling behavior hypothesis and highlights the strong coupling nature, assuming that the density of states at the Fermi level for Ag and Cu nanoparticles is equal to that of Pb.

I would like to suggest the publication of this manuscript. However, the authors should explain why they just used fixed values of Coulomb pseudopotentials. 

Author Response

My reply is in red

This work shows that the experimental critical temperature of nanoparticle systems, including superconducting Pb and normal Ag, Cu, and Al materials, is governed by the proximity effect and can be accurately explained using Eliashberg's theory. It refutes the initial weak coupling behavior hypothesis and highlights the strong coupling nature, assuming that the density of states at the Fermi level for Ag and Cu nanoparticles is equal to that of Pb.
I would like to suggest the publication of this manuscript. 

I thank the referee for his observations to which I will try to reply.

However, the authors should explain why they just used fixed values of Coulomb pseudopotentials. 

If the Coulomb potential could vary I would have one more free parameter in the theory and furthermore even by varying the Coulomb pseudopotential it is not possible to reproduce the experimental data in any way as 
I have been able to verify.

All these considerations have been included in the paper.

Round 2

Reviewer 1 Report

The Author has replied to my questions and I believe that the paper is almost ready for publication. I have, however, some final points that need to be corrected.

A)  On page 4 there is a sentence in italic that is not intelligible. It appears to be a sort of repetition of the sentence that follows it. So it is probably a mistake. Please fix it or remove it. 

B) The answer to the question whether the A parameter is relevant or not is not clear. 

I see three possible cases 

1] the parameter A enters the equations and is relevant.

2] the parameter A enters the equations but is almost irrelevant  apart from extreme (nonphysical cases) 

3] the parameter A doesn't enter the equations. 

From the reply to me I understand that the situation is that of case 2]. The Author says: 

we have verified that in a very wide range of values of A the final result is independent of the aforementioned  value. Only in extreme cases numerical problems could arise which do not allow obtaining a solution of Eliashberg's equations    

I'm ok with the statement as it sounds quite reasonable. 

But the statement seems to be at odds with the other statement, that the equations can be rewritten without any dependence from A.  i.e. case 3]

This leaves me very confused. The issue should be clarified before a final acceptance. If the Authors still wants to maintain the sentence about the possibility of rewriting the equations without the A parameter, probably the alternative formulation should be included and discussed. 

I recommend the paper to be published as soon as these last issues have been corrected

 

Author Response

I thank the referee for his observations that help me to clarify my work
I will try to reply to these observations.

The Author has replied to my questions and I believe that the paper is almost ready for publication. 
I have, however, some final points that need to be corrected.

A)  On page 4 there is a sentence in italic that is not intelligible. 
It appears to be a sort of repetition of the sentence that follows it. So it is probably a mistake. Please fix it or remove it. 

Yes there is some repetition because I wanted to be clear
I change the sentence in this way:

"We assume that, to explain the experimental data, it is necessary that the density of states at the Fermi level of the
normal metal nanoparticles has to be substitute, in the equations, by the value of the superconductor density of states
but only when the size of normal metal nanoparticles is approximately equal or minor to that of the superconductor coherence length."

B) The answer to the question whether the A parameter is relevant or not is not clear. 
I see three possible cases 

1] the parameter A enters the equations and is relevant.

2] the parameter A enters the equations but is almost irrelevant  apart from extreme (nonphysical cases) 

3] the parameter A doesn't enter the equations. 

From the reply to me I understand that the situation is that of case 2]. The Author says: 

we have verified that in a very wide range of values of A the final result is independent of the aforementioned  value. Only in extreme cases numerical problems could arise which do not allow obtaining a solution of Eliashberg's equations    

I'm ok with the statement as it sounds quite reasonable. 

But the statement seems to be at odds with the other statement, that the equations can be rewritten without any dependence from A.  i.e. case 3]

This leaves me very confused. The issue should be clarified before a final acceptance. If the Authors still wants to maintain the sentence about the possibility of rewriting the equations without the A parameter, probably the alternative formulation should be included and discussed. 

I recommend the paper to be published as soon as these last issues have been corrected

Yes the correct situation is that of case 2. I am sorry, I was not clear in my explanation and only added one more complication. 
I have removed the statement, that the equations can be rewritten without any dependence from A. 
I meant that equations can be written without using A but other quantities which however are a function of A 
so the problem remains and furthermore in this second way the physical interpretation is less clear. 
The best thing is to remove this sentence like I did.