In its simplest form BCS theory gives the superconducting transition temperature T_c in terms of electron-phonon coupling constant \lambda, Debye cutoff energy E_D and the density of states at the Fermi level N(0) by

k_BT_c = 1.13 E_D exp(-1/N(0))\lambda

Here there is no Fermi energy involved but just the density of states at the Fermi level. Why do you think that Fermi energy rather than density of states at the Fermi level should be correlated with T_c?

Of course the above simple BCS formula may not be appropriate for layered high-Tc materials for optimum doping. There is an interesting article (J. Phys.: Condens. Matter 23 (2011) 295701) which claims that optimum T_c called T_co can be given by

k_B T_co = \beta/(l\zeta)

where l is realated to the mean spacing between interacting charges between the layers, \zeta is the distance between the interacting layer and \beta is an universal constant. By this simple formula one can describe the optimum transition temperatures of 31 layer high-Tc superconductors of the classes of cuprate, ruthenates, ruthenocuprates, iron pnictides and organic superconductors. It seems that the Coulomb coupling between electronic bands in adjacent, spatially separated layers is the deciding factor along with their layered crystal structure, bond lengths, and valency properties of the ions.

In the attached figure you will see the correlation between T-co and other parameters.

Thank you Tapan Sir for your valuable answer....From theropower measurement, I have calculated Fermi temperature using relation S/T(0 K) = pi2*KB/(2eTF) applicable for metal at lower Tc.. where S is thermo power, KB Boltzmann constant and TF is the Fermi temperature. From above relation I am getting TF directly that's why I am connecting Fermi temperature with Tc. You may be also right that may be due to the variation in density of state. Then please tell me If you know...How can I connect TF with density of state?

Measure thermopower as a function of temperature starting from low temperature and go up to at least room temperature. Plot the data as a function of temperature and then try to fit the data with the equation you give and check whether the equation gives a satisfactory fit to your data. Your equation (see Ashcroft and Mermin, Solid State Physics, p. 52) is based on basically the free-electron theory albiet with Fermi statistics. It may be a fairly good approximation in the normal state however. Do this and let me know.

Thermopower has been measured on many superconducting materials. To fit the data it was necessary to add different correction terms to the equation. Also for ceramic samples other extra terms must be added. So unless you have well characterized single crystals it may not be very easy to fit the data.

There is a relationship in the following manner: the superconducting coherence length depends on the Fermi velocity (similar to the thermopower -- actually, the latter depends on the Fermi-surface average of the Fermi velocity, a fact that is irrelevant for isotropic models), and thereby on the Fermi temperature. Some care is however required in relating the Fermi energy (or the Fermi temperature) to the Fermi velocity, since only in the simplest model where ε(k) = ||k||^2/2 (in the atomic units) has one v_F = k_F = 2 ε_F/k_F.

As Tapan Chatterji has stated , the superconducting transition temperature is related to only with the DOS measured at the Fermi energy .Fermi temperature is not related to the superconducting transition temperature. We have done lot of calculations on superconducting transition temperature . One can also see our papers.

The graph reproduced by Tapan Chatterji is very interesting . It may be difficult to give any explanation .But, it works.

I found a correlation between electrongetivity and superconductivity .It seems to work. The explanation is missing.

In BCS theory no. But if you are talking about a superconductor not described by the BCS theory there is a Uemura plot which empirically links Tf and Tc in heavy fermions, cuprates etc... Additionally if you are close to another phase then correlations will be enhanced and this will decrease Tf. Finally if you measure thermopower and use it to directly access the fermi temperature you have to be very careful as firstly in a multiband system the thermopower is sensitive to all bands weighted by the conductivity and secondly you need to be in the T->0 limit to use that relation which you won't be as presumably you have to measure above Tc.

In the Bose-Einstein limit of dilute mobile charges and strong electron-electron attraction, the critical temperature is not given by the BCS formula but by the Bose-Einstein condensation temperature, given by k_B T_c approx. = 3.315 \hbar^2/m_B n_B^(2/3) where n_B is the density of preformed pairs and m_B their mass. If you take m_B=2 m_e (the mass of two electrons) and n_B = n/2 (the electron number density, divided by 2) then of course the density can be re-expressed in terms of the Femri energy.

please read "heavy fermions systems and high temperature superconductors: some common features in Journal of magnetism and magnetic materials" vol 76&77 (1988) p552-560, R. Tournier, A Sulpice , P. Lejay, O. Laborde, J. Beille