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 T c 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 Δ s ; 3) transforms the problem of optimizing j 0 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.
We trace in Section 2 the backdrop of Multi-Band Approach (MBA) for hetero-structured, multi-gapped superconductors (SCs) based on numerous papers, for the gist of which [
At the root of MBA is the work of Suhl et al. [
Indeed, numerous SCs have been listed in [
A striking feature of all SCs that have Tcs greater than that of Nb (≈9 K) is that they are multi-component materials, suggesting naturally that Cooper pairs (CPs) in them may also be bound via simultaneous exchanges of phonons with more than one species of ions. It has been shown [
Employing
I ( p ) = − ( V 1 + V 2 ) / ( 2 π 3 ) , ( ℏ = c = 1; V i = 0,except when E F − k θ i ≤ p 2 / 2 m * ≤ E F + k θ i ) as the kernel of a BSE, the following EF-incorporated equations have been derived for |W20| (to be identified with ∆2 > ∆1) and Tc, where ∆1 and ∆2 are any two gap-values of an SC which may also be characterized by additional ∆-values [
R e { λ 1 2 ∫ − k θ 1 k θ 1 d ξ ξ + μ | ξ | + | W 20 | / 2 + ( | W 20 | ; λ 1 → λ 2 ; θ 1 → θ 2 ) } = R e { [ ( μ − k θ 2 ) 3 / 2 + 3 4 ∫ − k θ 2 k θ 2 d ξ ξ + μ ( 1 − ξ ξ 2 + W 20 2 ) ] 1 / 3 } (1)
R e { λ 1 2 ∫ − k θ 1 k θ 1 d ξ ξ + μ tanh ( ξ / 2 k T c ) ξ + ( T c ; λ 1 → λ 2 ; θ 1 → θ 2 ) } = R e { [ 3 4 ∫ − μ k θ 2 d ξ ξ + μ { 1 − tanh ( ξ / 2 k T c ) } ] 1 / 3 } , (2)
where chemical potential μ has been used interchangeably with EF, θ1 and θ2 > θ1 are the Debye temperatures of the ion-species that cause pairing and λ 1 ≡ [ N ( 0 ) V 1 ] and λ 2 ≡ [ N ( 0 ) V 2 ] their interaction parameters, no distinction is made between the values of μ and the λs at T = 0 and T = Tc, and Re ensures that the integrals yield real values even when μ < kθ2. Note that when λ2 = 0, λ1 = λ, θ1 = θ, |W20| = |W| and μ ≫ k θ , (2) becomes identical with the BCS equation for Tc of an elemental SC, and (1) leads to | W | = 2 k θ / [ exp ( 1 / λ ) − 1 ] ( Δ = k θ / sinh ( 1 / λ ) ) , where in the parentheses is noted the BCS equation for ∆. Via a detailed comparative study of these equations for six elemental SCs [
It has been shown that [
s ( E F ) ≡ m * / m e = A 1 ( γ / v g ) 2 / 3 E F − 1 / 3 ,
n s ( E F ) = A 2 ( γ / v g ) E F ,
P 0 ( E F ) = A 3 ( θ / y ) ( γ / v g ) 1 / 3 E F − 2 / 3 , and
v 0 ( E F ) = A 4 ( θ / y ) ( γ / v g ) − 1 / 3 E F − 1 / 3 hence, from the definition,
j 0 = ( n s / 2 ) e * v 0 ( v 0 = P 0 / 2 m * ) , it follows that [
j 0 = A 5 ( θ / y ) ( γ / v g ) 2 / 3 E F 2 / 3 ( A 5 = 6.146 × 10 − 4 C ⋅ eV − 4 / 3 ⋅ K 1 / 3 ⋅ s − 1 ) , (3)
where
A 1 ≃ 3.305 × 10 − 10 eV − 1 / 3 ⋅ cm 2 ⋅ K 4 / 3 ,
A 2 ≃ 2.729 × 10 7 eV − 2 ⋅ K 2 ,
A 3 ≃ 1.584 × 10 − 6 eV 4 / 3 ⋅ cm ⋅ K − 1 / 3 , and
A 4 ≃ 1.406 × 10 8 eV 2 / 3 ⋅ sec − 1 ⋅ K − 5 / 3 .
In these equations, θ is the Debye temperature of the SC and θ1 and θ2 the Debye temperatures of ion-species that cause pairing, m* (me) is the effective (free) electron mass, γ the electronic specific heat constant and vg the gram-atomic volume of the SC; (ns/2), e* (twice the electronic charge), v0, and P0 are, respectively, the number density, electronic charge, critical velocity and critical momentum of CPs (momentum at which ∆ vanishes), and y = ( k θ / P 0 ) 2 m * / E F , a dimensionless construct to be obtained by solving 1 ≅ λ 1 [ r 1 y ln { r 1 y / ( r 1 y − 1 ) } + ln ( r 1 y − 1 ) ] + ( λ 1 → λ 2 , r 1 → r 2 ) , where r i = θ i / θ .
This equation is derived via a BSE with the same kernel as employed for (1) and (2), except that now
V i = 0 , unless E F − k θ i ≤ ( P 0 / 2 + p ) 2 / 2 m * , ( P 0 / 2 − q ) 2 / 2 m * ≤ E F + k θ i .
A more accurate (but rather elaborate) equation that additionally contains EF explicitly has been derived in [
1) Tl2Ba2CaCu2O8 (Tl-2212) [
2) Ba0.6K0.4Fe2As2 (BaAs) [
s = 0.420 , n s = 3.1 × 10 20 ( cm − 3 ) , v 0 = 50 × 10 4 cm / sec , ξ = 7 ( Å ) .
1) CA satisfies the criteria noted for MBA in Section 2 because: (1) and (2) hold for arbitrary values of EF, the ions responsible for pairing have been distinguished by distinct θ- and λ-values and the valence band overlapping the undulating Fermi surface has been characterized by locally spherical values―reminiscent of the locally inertial frames employed in the general theory of relativity [
2) A salient feature of CA is that it invariably appeals to the ion-species that comprise an SC, whereas the number of bands invoked in MBA for the same SC differs from author to author [
3) While (3) identifies the parameters that can enhance j0, and hence Tc [
4) To conclude, with s±-wave as an intrinsic feature of it, we have shown that CA transforms the problem of raising Tc into one of chemical engineering and that it is applicable to a wide variety of SCs, including the Fe-based SCs―without invoking a new state for them, as has been suggested via MBA [
The author thanks D. C. Mattis, D. M. Eagles, A. Bianconi, R. Hott and A. Semenov for valuable correspondence, and L. K. Pande for a critical reading of the manuscript.
Malik, G.P. (2018) An Overview of the Multi-Band and the Generalized BCS Equations-Based Approaches to Deal with Hetero-Structured Superconductors. Open Journal of Microphysics, 8, 7-13. https://doi.org/10.4236/ojm.2018.82002