Narrow Results

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 ≤ T_c 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-T_c 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 T_c uniquely. Applications of these equations will be taken up in a subsequent paper.

Based on the concepts of a superpropagator, multiple Debye temperatures, and equivalence of the binding energy of a Cooper pair and the BCS energy gap, the set of generalized BCS equations obtained recently via a temperature-generalized Bethe–Salpeter equation is employed for a unified study of the following composite superconductors: MgB₂, Nb₃Sn, and YBCO. In addition, we study the Nb–Al system in which Cooper pairs as resonances have recently been reported to have been observed. Our approach seems to suggest that a simple extension of the BCS theory that accommodates the concept of Cooper pairs bound via a more than one phonon exchange mechanism may be an interesting candidate for dealing with high-temperature superconductors.