Employing theGeilikman-Kresin (GK) theory, we address the experimental data obtained by Bauer et al., and by Schneider et al.,on the thermal conductivity (κ) of superconducting MgB_{2}. The two gaps of this compound have qualitatively been understood via the well-known Suhl, Matthias, and Walker’s (SMW) approach to multigap superconductivity. Since this approach is based on one-phonon exchange mechanism for the formation of Cooper pairs, it cannot give a quantitative account of the values of T_{c} and the multiple gaps that characterize MgB_{2} and other high-T_{c} superconductors (SCs). Despite this fact and some rather ambiguous features, it has been pointed out in a recent critical review by Malik and Llano (ML) that the SMW approach provides an important clue to deal with an SC the two gaps of which close at the same T_{c}: consider the possibility of the interaction parameters in the theory to be temperature-dependent. Guided by this clue, ML gave a

Recasting the BCS theory in the larger framework of the Bethe-Salpeter equation, a new equation is derived for the temperature-dependent critical current density j_{c}(T) of an elemental superconductor (SC) directly in terms of the basic parameters of the theory, namely the dimensionless coupling constant [N(0)V], the Debye temperature θ_{D} and, additionally, the Fermi energy E_{F}—unlike earlier such equations based on diverse, indirect criteria. Our approach provides an ab initio theoretical justification for one of the latter, text book equations invoked at T = 0 which involves Fermi momentum; additionally, it relates j_{c} with the relevant parameters of the problem at T ≠ 0. Noting that the numerical value of E_{F} of a high-T_{c} SC is a necessary input for the construction of its Fermi surface—which sheds light on its gap-structure, we also briefly discuss extension of our approach for such SCs.

Abstract:
Heavy-fermion superconductors (HFSCs) are regarded as outside the purview of BCS theory because it is usually constrained by the inequality , where E_{F}, μ, k_{B}, and θ_{D} are, respectively, the Fermi energy, chemical potential, Boltzmann constant, and the Debye temperature. We show that this restriction can be removed by incorporating μ into the equations for T_{c} and the gap Δ_{0} at T = 0. Further, when μ < k_{B}θ_{D}, we curtail the limits of the equations for T_{c} and Δ_{0} to avoid complex-valued solutions. The resulting equations are applied to a prominent member of the HFSC family, i.e., CeCoIn_{5}, by appealing to ideas due to Born and Karmann, Suhl et al., and Bianconi et al. Since the equations now contain an additional variable μ, we find that 1) the T_{c} of the SC can be accounted for by a multitude of values of the (μ, λ) pair, λ being the interaction parameter; 2) the λ vs. μ plot has a dome-like structure when μ < k_{B}θ_{D}; 3) the (μ, λ) values obtained in 2) lead to reasonable results for the range of each of the following variables: Δ_{0}, s, and n, where s is the ratio of the mass of a conduction electron and the free electron mass and n is the number density of charge carriers in the SC.

Abstract:
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 (j_{0}) 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 T_{c} and Δs ; 3) transforms the problem of optimizing j_{0} of a composite SC, and hence its T_{c}, 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.

Abstract:
By generalizing the isotope effect for elemental superconductors (SCs) to the case of pairing in the 2-phonon exchange mechanism for composite SCs, we give here an explanation of the well-known increase in the critical temperature (T_{c}) of Bi_{2}Sr_{2}CaCu_{2}O_{8} from 95 K to 110 K and of Bi_{2}Sr_{2}Ca_{2}Cu_{3}O_{10} from 105 to 115 - 125 K when Bi and Sr in these are replaced by Tl and Ba, respectively. On this basis, we also give the estimated T_{c}s of some hypothetical SCs, assuming that they may be fabricated by substitutions similar to Bi → Tl and Sr → Ba.

Abstract:
We address the T_{c} (s) and multiple gaps of La_{2}CuO_{4} (LCO) via generalized BCS equations incorporating chemical potential. Appealing to the structure of the unit cell of LCO, which comprises sub- lattices with LaO and OLa layers and brings into play two Debye temperatures, the concept of itinerancy of electrons, and an insight provided by Tacon et al.’s recent experimental work concerned with YBa_{2}Cu_{3}O_{6.6} which reveals that very large electron-phonon coupling can occur in a very narrow region of phonon wavelengths, we are enabled to account for all values of its gap-to-T_{c} ratio (2Δ_{0}/k_{B}T_{c}), i.e., 4.3, 7.1, ≈8 and 9.3, which were reported by Bednorz and Müller in their Nobel lecture. Our study predicts carrier concentrations corresponding to these gap values to lie in the range 1.3 × 10^{21} - 5.6 × 10^{21} cm^{-3}, and values of 0.27 - 0.29 and 1.12 for the gap-to-T_{c} ratios of the smaller gaps.

Abstract:
The recent concern with the role of Fermi energy (E_{F}) as a determinant of the properties of a superconductor (SC) led us to present new E_{F}-dependent equations for the effective mass (m*) of superconducting electrons, their critical velocity, number density, and critical current density, and also the results of the calculations of these parameters for six SCs the T_{c}s of which vary between 3.72 and 110 K. While this work was based on, besides an idea due to Pines, equations for Tc and the gap at T = 0 that are explicitly E_{F}-dependent, it employed an equation for the dimensionless construct that depends on E_{F} only implicitly; k in this equation is the Boltzmann constant, θ is the Debye temperature, and P0 is the critical momentum of Cooper pairs. To meet the demand of consistency, we give here derivation of an equation for y that is also explicitly E_{F}-dependent. The resulting framework is employed to (a) review the previous results for the six SCs noted above and (b) carry out a study of NbN which is the simplest composite SC that can shed further light on our approach. The study of NbN is woven around the primary data of Semenov et al. For the additional required inputs, we appeal to the empirical data of Roedhammer et al. and of Antonova et al.

Abstract:
Guided by the belief that Fermi energy E_{F} (equivalently,
chemical potential μ) plays a pivotal？role in determining the
properties of superconductors (SCs), we have recently derived μ-incorporated
Generalized-Bardeen-Cooper-Schrieffer？equations (GBCSEs) for the gaps (Δs)
and critical temperatures (T_{c}s) of both elemental and
composite SCs. The μ-dependent interaction parameters consistent
with the values of Δs and T_{c}s of any of these SCs were
shown to lead to expressions for the effective mass of electrons (m^{*})
and their number density (n_{s}), critical velocity (v_{0}),
and the critical current density j_{0} at T = 0 in
terms of the following five parameters: Debye temperature, E_{F},
a dimensionless construct y, the specific heat constant, and the gram-atomic
volume. We could then fix the value of μ in any SC by
appealing to the experimental value of its j_{0} and
calculate the other parameters. This approach was

Abstract:
We develop a $GL_{qp}(2)$ invariant differential calculus on a two-dimensional noncommutative quantum space. Here the co-ordinate space for the exterior quantum plane is spanned by the differentials that are commutative (bosonic) in nature.