Some Implications of an Alternate Equation for the BCS Energy Gap

Gulshan Prakash Malik, Manuel de Llano

Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México Apdo, México City, México.
Theory Group, School of Environmental Sciences, Jawaharlal Nehru University, New Delhi, India.

DOI: 10.4236/jmp.2013.44A002                

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_cs and the multiple gaps of a variety of high-temperature superconductors (SCs). These equations are formulated in terms of the binding energies W₁(T),W₂(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 |W₀₁|=△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|W₀₁|. In this note we carry out a detailed comparison between the GBCSE-based W₁(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 W₁(T) provides a viable alternative to the BCS equation for △(T). This suggests the use of, when required, the equation for W₂(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.

Keywords

BCS and Generalized-BCS Equations; T-Dependent Equations for BCS Gap and for Binding Energy of Cooper Pairs; Elemental and Composite Superconductors

Conflicts of Interest

The authors declare no conflicts of interest.

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