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The BCS theory predicts that the critical temperature of the superconducting transition is given by

$$ T_c \approx \theta \exp \left (- \frac{1}{U D(\epsilon_F)} \right ) $$

where $\theta$ is the Debye temperature, $U$ is the coupling constant of the electron-phonon interaction and $D(\epsilon_F)$ the density of states at the Fermi level.

In other words, the larger $D(\epsilon_F)$ (or $U$) is the higher the critical temperature gets. However, strictly speaking, this relation is true only in the framework of BCS and in the weak coupling approximation (i.e. $U D(\epsilon_F) \ll 1$).

Question: is there any experimental (or even theoretical) evidence that the rule "higher DOS at the Fermi level implies higher $T_c$" holds also for non-BCS superconductors or outside the weak coupling regime? I'm not asking specifically about the exponential rule, but about the general dependence of $T_c$ on $D(\epsilon_F)$. I'd appreciate any references if this is indeed the case.

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I think it is misleading to try to separate the DOS and the interaction strength. Rather one should think of $UD(\epsilon_F)$ as a dimensionless coupling strength. This view is further reinforced, if one looks at the problem from the point of view of the renormalization group, where reducing the energy cutoff (i.e., changing the DOS, shrinking the band to a thin strip around the Fermi level) has to be compensated by adjusting the coupling strength, whereas their product at the stable point remains unchanged.

This however does not yet answer your question, since applying the renormalization group means using an effective field theory, valid here only in the weak coupling regime. The point here is that outside of this regime we may either be unable to build an effective theory for the phase transition (which confines us to using ampirical description) or we will have totally different form of the effective coupling parameter and a different theory associated with it. In any case, it will not be anymore an effective theory built in the viscinity of the Fermi energy, and will not be expressible in terms of the values of the parameters (including the DOS) at this energy.

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  • $\begingroup$ Thank you for the explanation. But can't we actually measure this dimensionless coupling parameter in experiments and see how it affects the critical temperature? If I understand correctly, your answer mainly addresses the theoretical framework (or lack thereof) outside the weak coupling regime. $\endgroup$
    – grjj3
    Commented May 29, 2021 at 19:15
  • $\begingroup$ I suppose that it is the only thing that can be measured experimentally, rather than separately $U$ and DOS. Also, we cannot really modify the coupling or DOS independently - but we compare different materials. $\endgroup$
    – Roger V.
    Commented May 29, 2021 at 19:22
  • $\begingroup$ And do the experiments indicate that bigger coupling parameter leads to higher critical temperature? $\endgroup$
    – grjj3
    Commented May 29, 2021 at 19:53
  • $\begingroup$ That I can't say: superconductivity is not my field - I was naking a rather general point iny answer. But it is worth checking in Tinkham's book. $\endgroup$
    – Roger V.
    Commented May 29, 2021 at 20:02
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It is an experimental fact that HTS superconductors generally show large temperature-independent Pauli susceptibility, indicating high DOS near the Fermi surface. See e.g. here.

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