Dear Mohamed Lamine Tounsi, as you state in the formulation of this thread.

All this is could be due to Fermi level positions of the electron that form the cooper pair in the wave vector space, or maybe to energy bands or both or other features.

However, In 1950, Maxwell and Reynolds et al. found that the critical temperature of a superconductor depends on the isotopic mass of the constituent element [1][2]

This important discovery pointed to the electron-phonon interaction as the microscopic mechanism responsible for superconductivity.

But for other compounds, nobody knows, only the BCS theory is the one adopted microscopically, and it works mainly for metals and some binary and ternary compounds at low temperatures.

Otherwise, the Ginzburg Landau phenomenological theory works pretty well in many other cases.

Let's see what researchers working on high Tc and other unconventional superconductors have to say about your inquiry. Thank you.

[1] E. Maxwell (1950). "Isotope Effect in the Superconductivity of Mercury". Physical Review. 78 (4): 477.

[2] C. A. Reynolds; B. Serin; W. H. Wright & L. B. Nesbitt (1950). "Superconductivity of Isotopes of Mercury". Physical Review. 78 (4): 487.

https://en.wikipedia.org/wiki/Superconductivity

Dear Mohamed Lamine Tounsi: Heavy fermion materials are a specific type of intermetallic ompound, containing elements with 4f or 5f electrons in unfilled electron bands. I have already underlined in your previous question “How to achieve superconductivity at the highest temperature?” we know since almost sixty years that the xenon and some other gas call “inert” has valence and bonds properties see “Chernick C.L., Rec. Chem. Prog., 24, 139-155, (1962)”. As a result, in superconductors as La2-xBaxCuO4 where the bivalent barium replace the trivalent lanthanum it is possible to consider that the xenon shell of Ba or La contributes to the valance, see “From Ln valence to that of 3d”.

The answer to your new question have to take into account the chemical bonds resulting of the xenon shell for the heavy fermion materials or similar bonds with the 3d metals.

Now for the statistical aspects of the electrical conduction there is a missing point the classical statistical function have introduce the balance between the particles with high energy and those with low energy. In other words, there are many poor people for one rich people. See “About the statistical distribution of the thermal energy”.

Dear Prof. Xavier Oudet, very interesting. Could that reasoning be applied for example to the interplay been magnetism and superconductivity observed in Upt3?

Dear Mohamed,

different Tc emerge from two points:

1. different pairing energy of electrons. That is the energy advantage from formation of singlet pairs.

2. different position of the Fermi energy (Ef) to a Brillouin zone edge. The most optimal case is that Ef leans exactly on a Brillouin zone edge. In noble metals Ef is much higher, so SC is impossible. In insulators Ef is much lower - no SC. Therefore most superconductors are bad conducting metals.

Dear Professor P. Contreras: Many thanks for your question. I spent about ten years, 1965-1975, in the Rare Earths Lab. in Bellevue France, so I know about the Ln but not too much about the Actinides. Now, the Radon shell of these elements is certainly involved in the chemical bonds of their compounds, think for example to UF6, which lead to think to XeF6 and son, see “Fluorine Compounds of Xenon and Radon” C.E. Chernik et all that is 17, authors, 137-138, Science Vol.138, 1962. You will understand better the chemical bonds of these elements with this approach. For Pt You can also invoke the Xenon shell, like for the 3d metals see “From Ln valence to that of 3d”.

Thank you so much for your kind and detailed answer, Dear Prof. Xavier Oudet. I will read the publication.

I always have been amazed by the unusual superconducting properties of inner elements, lets say for example the 3He phases. And although 3He does not have so many electrons as the Radon or the Xenon shells do, all they involve very unusual and exotic properties in superconductors. 3d metals also have unusual superconducting properties, Sr2RuO4 for example.

The electron-phonon coupling and the energy range over which it is effective are the two reasons which determine the superconducting transition temperature TC. The Debye energy in solids can be considered as the energy range.

The inclusion of Coulomb repulsion makes an improvement on the elementary theory with an additional parameter known as Coulomb pseudopotential (mu*).

What is important for understanding superconductivity is the mechanism that leads to the appearance of a critical temperature at all. And once that's understood, it follows that its value depends, also, on properties specific to each material. For applications these specific features, of course, are of interest, once the general framework's understood.

Interactions are ``weak'' or ``strong'' compared to some scale-and it's how that scale appears, that's of relevance.

So the key idea for how such a scale can appear is the competition between the Coulomb repulsion of conduction electrons and the attraction between them, that's mediated by the elastic deformations of the material.

When repulsion wins, the material is a conductor; when attraction wins, the material is a superconductor. And determining which one wins under what circumstances is controlled by the properties of the material and how these can be affected by outside agents.

There's a lot more that's known, of course, and recent experiments have led to this: https://www.quantamagazine.org/physicists-discover-first-room-temperature-superconductor-20201014/ Which illustrates even more strongly the fact that pressure plays a crucial role in raising the transition temperature.