^{*}

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
*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.

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. [_{c}s arise in this approach because the BCS interaction parameter λ ≡ [N(0)V] is now determined not via a single interaction energy “V”, but via a quadratic equation involving three such energies: V_{s} and V_{d} due to scattering in the two bands individually and V_{sd} due to cross-band scattering. Since in this model the equation employed to determine T_{c}―for each value of λ―is the familiar BCS equation for elemental SCs derived for one-band, weak-coupling (λ < 0.5) theory, it cannot per se explain the occurrence of high-T_{c}s. For this reason, the multi-band concept is supplemented by the well-known Migdal-Eliashberg-McMillan approach [_{e}/M), where m_{e} is the mass of an electron and M that of an ion. MBA has evolved around these basic ideas because anisotropic SCs necessitate that [_{F} = Fermi energy, k = Boltzmann constant; θ = Debye temperature) be abandoned; 2) different locations in k-space should be characterized by distinct pairing strengths and order parameters (i.e., gaps); and 3) the assumption that the Fermi surface is isotropic/spherical be dispensed with.

Indeed, numerous SCs have been listed in [

A striking feature of all SCs that have T_{c}s 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 [_{c} of an elemental SC can also be obtained via a Bethe-Salpeter equation (BSE) with a kernel corresponding to the one-phonon exchange mechanism (1PEM) in the ladder approximation. The first diagram in this series has one rung, the second two rungs, and so on. If the number of rungs between any two space-time points in each of these diagrams is doubled, then we have the 2-phonon exchange mechanism (2PEM) in operation. Similarly―depending on composition of the SC―CPs may also be bound via a 3-phonon exchange mechanism (3PEM). It hence follows that in a composite SC, CPs may exist with different values of the binding energy (2|W|). Since the inequalities | W 3PEM | > | W 2PEM | > | W 1PEM | must hold, and since |W| = ∆ [

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 E_{F}-incorporated equations have been derived for |W_{20}| (to be identified with ∆_{2} > ∆_{1}) and T_{c}, 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 E_{F}, θ_{1} and θ_{2} > θ_{1} 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 = T_{c}, and Re ensures that the integrals yield real values even when μ < kθ_{2}. Note that when λ_{2} = 0, λ_{1} = λ, θ_{1} = θ, |W_{20}| = |W| and μ ≫ k θ , (2) becomes identical with the BCS equation for T_{c} 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 [^{±}-wave feature is an inbuilt feature of (1) since it has been derived by assuming that the signature of W_{20} changes on crossing the Fermi surface. Such an assumption leaves the BCS equation for ∆ unchanged because it is quadratic in ∆.

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 θ_{1} and θ_{2} the Debye temperatures of ion-species that cause pairing, m* (m_{e}) is the effective (free) electron mass, γ the electronic specific heat constant and v_{g} the gram-atomic volume of the SC; (n_{s}/2), e* (twice the electronic charge), v_{0}, and P_{0} 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 E_{F} explicitly has been derived in [

1) Tl_{2}Ba_{2}CaCu_{2}O_{8} (Tl-2212) [_{Ca} = 254 K, θ_{Tl} = 289 K, θ_{Ba} = 296 K. Assuming that pairing is due to the Ca and Tl ions and treating μ as an independent variable, we find via (1) and (2) that the empirical values |W_{20}| = 23.9 meV and T_{c} = 110 K of the SC can be explained by a multitude of {μ, λ_{Ca}, λ_{Tl}}-values. For each such set of values, we solve the equation for y given above and then calculate j_{0} via (3), which leads to a multitude of values for the larger set {μ, λ_{Ca}, λ_{Tl}, y, j_{0}}. Appeal to the empirical value, j_{0} = 6.5 × 10^{7} A/cm^{2}, then fixes the set as: {μ = 49.8 meV, λ_{Ca} = 0.4899, λ_{Tl} = 0.4543, y = 1.512}. As predictions, we have: s = 4.23, n_{s} = 4.39 × 10^{20} cm^{−}^{3}, v_{0} = 9.35 × 10^{5} cm/sec. Repetition of the above procedure for pairing via the Ca and Ba ions, or the Ba and Tl ions, fixes λ_{Ba}. Each of the sets {θ_{Ca}, λ_{Ca}}, {θ_{B}_{a}, λ_{B}_{a}} and {θ_{T1}, λ_{T1}} then leads to a value of |W_{10}| in the 1PEM scenario, and to |W_{30}| in the 3PEM scenario when all these sets operative simultaneously.

2) Ba_{0.6}K_{0.4}Fe_{2}As_{2} (BaAs) [_{Ba} = 124.6 K, θ_{Fe} = 399.4 K and θ_{As} = 148.6 K from θ_{BaAs} = 274 K. From among the multitude of empirical gap-values that characterize it, , e.g., ≈ 0, 2.5, 3.3, 3.6, 4, 6, 7, 7.6, 8.5, 9, 12 meV, we choose 6 and 12 meV as our starting point and take its T_{c} as 37 K, which are also the values commonly employed in MBA. We now assume that while the smaller gap and the T_{c} are due to the Ba and Fe ions, the larger gap pertains to the 3PEM scenario (involving also the As ions). This necessitates supplementing (1) and (2) by another equation, which follows from (1) by replacing |W_{20}| by |W_{30}| and adding ( | W 30 | ; λ 1 → λ 3 ; θ 1 → θ 3 = θ A s ) to its LHS; θ_{2} = θ_{Fe} on the RHS remains unchanged because it is greater than either of θ_{1} (=θ_{Ba}) and θ_{2} (=θ_{As}). By solving three simultaneous equations, viz., (1), (2), and (3), as against the normal practice of appealing only to the T_{c} and Δ-values of the SC, we are now led to a multitude of values for the set {μ, λ_{Ba}, λ_{Fe}, λ_{As}, y, j_{0}}. Appeal to j_{0} = 2.5 × 10^{7} Acm^{−}^{2} then fixes this set as {μ = 14.2 meV, λ_{Ba} = 0.1155, λ_{Fe} = 0.3838, λ_{As} = 0.2196, y = 3.433}. Besides, we are thus led to a quantitative explanation of several empirical features of the SC, such as: E_{F}/kT_{c} = 4.45, gap-values other than 6 and 12 meV, e.g., ≈ 0 and 9 meV, (T_{c})_{max} (via 3PEM) as exceeding 50 K, and the “dome-like” structure of its T_{c} vs. a tuneable variable; and, as predictions, to values of s, n_{s}, v_{0}, and coherence length ξ as:

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 E_{F}, 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}, YBCO, Bi-2212 and Tl-2212 [_{3}, La_{2}CuO_{4}, CeCoIn_{5} [_{0.6}K_{0.4}Fe_{2}As_{2} [

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 [_{c} and j_{0} under the same umbrella―which are features not shared by MBA.

3) While (3) identifies the parameters that can enhance j_{0}, and hence T_{c} [_{F}, so do its constituents m* and P_{0}. Besides, any attempt to increase the value of (γ/v_{g}), which is also implicitly a function of E_{F}, is bound to raise the problem of stability of the SC. Hence, in the quest for tangible clues to raise T_{c}s of SCs, we need to have a comprehensive catalogue that includes, besides their T_{c}- and ∆-values, the values of θ, j_{0}, m*, v_{0}, n_{s}, γ and v_{g}.

4) To conclude, with s^{±}-wave as an intrinsic feature of it, we have shown that CA transforms the problem of raising T_{c} 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