Some Implications of an Alternate Equation for the BCS Energy Gap

Abstract

A set of generalized-BCS equations (GBCSEs) was recently derived from a temperature-dependent Bethe-Salpeter equation and shown to deal satisfactorily with the experimental data comprising the T c s and the multiple gaps of a variety of high-temperature superconductors (SCs). These equations are formulated in terms of the binding energies W 1 ( T ), W 2 ( T ),… of Cooper pairs (CPs) bound via one- and more than one-phonon exchange mechanisms; they contain no direct reference to the gap/s of an SC. Applications of these equations so far were based on the observation that for elemental SCs | W 01 |= △ 0 at T = 0 inthe limit of the dimensionless BCS interaction parameter λ → 0 . Here △ 0 is the zero-temperature gap whence it follows that the binding energy of a CP bound via one-phonon exchanges at T = 0 is 2| W 01 | . In this note we carry out a detailed comparison between the GBCSE-based W 1 ( T ) and the BCS-based energy gap △ ( T ) for all 0 ≤ T ≤ T c and realistic, non-vanishingly-small values of λ . Our study is based on the experimental values of T c Debye temperature , and ? 0 of several selected elements including the “bad actors” such as Pb and Hg. It is thus established that the equation for W 1 ( T ) provides a viable alternative to the BCS equation for △ ( T ) . This suggests the use of, when required, the equation for W 2 ( T ) which refers to CPs bound via two-phonon exchanges, for the larger of the two T -dependent gaps of a non-elemental SC. These considerations naturally lead one to the concept of T -dependent interaction parameters in the theory of superconductivity. It is pointed out that such a concept is needed both in the well-known approach of Suhl et al. to multi-gap superconductivity and the approach provided by the GBCSEs. Attention is drawn to diverse fields where T -dependent Hamiltonians have been fruitfully employed in the past.