An Overview of the Multi-Band and the Generalized BCS Equations-Based Approaches to Deal with Hetero-Structured Superconductors

We trace the conceptual basis of the Multi-Band Approach (MBA) and recall the reasons for its wide following for composite superconductors (SCs). Attention is then drawn to a feature that MBA ignores: the possibility that electrons in such an SC may also be bound via simultaneous exchanges of quanta with more than one ion-species—a lacuna which is addressed by the Generalized BCS Equations (GBCSEs). Based on several papers, we give a concise account of how this approach: 1) despite employing a single band, meets the criteria satisfied by MBA because a) GBCSEs are derived from a temperature-incorporated Bethe-Salpeter Equation the kernel of which is taken to be a “superpropagator” for a composite SC-each ion-species of which is distinguished by its own Debye temperature and interaction parameter and b) the band overlapping the Fermi surface is allowed to be of variable width. GBCSEs so-obtained reduce to the usual equations for the Tc and Δ of an elemental SC in the limit superpropagator → 1-phonon propagator; 2) accommodates moving Cooper pairs and thereby extends the scope of the original BCS theory which restricts the Hamiltonian at the outset to terms that correspond to pairs having zero centre-of-mass momentum. One can now derive an equation for the critical current density (j0) of a composite SC at T = 0 in terms of the Debye temperatures of its ions and their interaction parameters— parameters that also determine its Tc and Δs ; 3) transforms the problem of optimizing j0 of a composite SC, and hence its Tc, into a problem of chemical engineering ; 4) provides a common canopy for most composite SCs, including those that are usually regarded as outside the purview of the BCS theory and have therefore been called “exceptional”, e.g., the heavy-fermion SCs; 5) incorporates s±-wave superconductivity as an in-built feature and can therefore deal with the iron-based SCs, and 6) leads to presumably verifiable predictions for the values of some relevant parameters, e.g., the effective mass of electrons, for the SCs for which it has been employed.