Employing the Bethe–Salpeter equation (BSE) and the Matsubara recipe, and invoking both the electron–electron and the hole–hole scattering channels, we establish that the binding energy (W) of a Cooper pair (CP) is real, and equals the BCS energy gap (Δ) for all T ≤ Tc for a one-component superconductor. Given that the BCS theory is a generalization of the Hartree–Fock theory (generalized to allow for particle number fluctuations), the cognescenti would expect this result as a direct consequence of Koopman's theorem, proved for and well-known in the latter theory. However, this theorem is seldom mentioned in the literature on superconductivity; on the contrary, there is the statement in well-known texts that the binding energy of a CP becomes imaginary when the above-stated scattering channels are invoked for their formation. The importance of |W| = |Δ| for high-Tc superconductors is brought out by replacing the one-particle propagator in the BSE by a superpropagator — a field-theoretic construct apt for dealing with composite superconductors (CSs). A set of generalized BCS equations is thus obtained which, with the input of the multiple gaps of a CS, enables one to calculate its Tc uniquely. Applications of these equations will be taken up in a subsequent paper.
A detailed quantitative study of the pnictide composite superconductor (CS) Ba0.6K0.4Fe2As2 is presented in the frame-work of the recently derived set of generalized BCS equations. Invoking multiple Debye temperatures to take into account anisotropy of the CS, we address the current experimental data on its Tc and the (not so clear-cut) gap-values via different theoretical scenarios that attempt to identify the ion species responsible for pairing in it. This is done with the aid of the Bogoliubov’s restriction on the BCS dimensionless electron-phonon coupling constant. Significantly, our study sheds light on the gaps which have recently been observed in different iron-pnictide CSs as nodes or line-nodes on the Fermi surface and have evinced considerable interest.
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