The gaps of a hetero-structured SCs are most widely studied via an adaptation of the multi-band approach (MBA) of Suhl et al. [4]. When such a study also deals with a T_c of the SC, which is not always the case, it generally appeals to the Migdal-Eliashberg-McMillan approach (MEMA) [5]. Even though MEMA is cast in the BCS mould of a 1-phonon exchange mechanism (1 PEM) for the formation of CPs, it is employed to deal with such high-T_cs as are observed because it permits the BCS interaction parameter λ to even exceed unity because it is based on an integral equation the expansion parameter of which is not λ, but m_e/M where m_e is the electron mass and M the mass of an ion. Insofar as j_c is concerned, a plethora of formulae is conventionally employed depending upon the type of the SC (I or II), its size, shape and the manner of preparation—for some such formulae which do not take into account the T-dependence of j_c, see [6].

Insofar as j_c(T) is concerned, it is empirically known that it has its maximum value at T = 0 and that j_c(T_c) = 0. It took considerable time for theoretical attempts to evolve before the observed variation of j_c(T) between these limits could be explained. Perhaps the earliest such attempt was due to London, who gave an equation for j_c(T) valid at T = T_c, but which failed at lower temperatures because, as was later realized, it did not take into account the effect of the change in the order parameter with current/temperature. The equation for j_c(T) given by the phenomenological Ginzberg-Landau (GL) theory marks the next stage in the said evolution. This equation works well close to T_c, but not for much lower temperatures. As is well known, the GL theory reduces to the London theory when the concentration of superconducting electrons is uniformly distributed and that, as was shown by Gor’kov, the microscopic theory in which the energy gap is taken as an order parameter leads to the GL theory near T_c. These brief considerations suggest the need to appeal to the microscopic theory in order to explain the observed variation of j_c(T) for all T ≤ T_c. Before we do so, it is relevant to draw attention to a phenomenological equation for j_c(T) given by Bardeen [2] post-BCS. This equation, valid for all T between 0 and T_c and obtained by treating the gap as a variational parameter and minimizing the free energy for a given current, is

j c ( T ) = 1 2 2 j c ( 0 ) [ 1 − ( T / T c ) 2 ] 3 / 2 . (1)

Equation (1) is applicable to SCs for which changes in the energy gap with position can be neglected.

Since the thin samples of Romijn et al. [1] satisfied the condition(s) of validity of (1), they applied it to one of their samples and found that it indeed fits their data well. Nonetheless, (1) does not address the core issue of the problem, i.e., to identify the parameters on which j_c(T) depends. The knowledge of these is essential because it provides a handle to control j_c(T). To unravel what lies beneath the “blanket” of (1), one needs a microscopic theory, viz. the theory given by Eilenberger which is derived from the original Gor’kov theory under the assumption ρ F l ≫ 1 , where ρ_F is the electrical resistivity of electrons at the Fermi surface and l their mean free path. It was shown by KL [3] that for an SC subject to certain constraints, the Eilenberger equations can be further simplified. Since their samples satisfied these conditions, Romijn et al. [1] also employed approximate solutions of the KL equations and found that their data were thus adequately explained.

Complementing MBA and presented in a recent monograph [7] is an approach based on the GBCSEs which too has been applied to a significant number of SCs. One of the premises of this approach is that Fermi energy (E_F) plays a fundamental role in determining the superconducting properties of an SC. We recall in this connection that the usual BCS equations for the T_c and Δ of an elemental SC are independent of E_F because of the assumption that E F ≫ k θ , where k is the Boltzmann constant and θ the Debye temperature of the SC. GBCSEs are obtained via the Matsubara technique and a Bethe-Salpeter equation (BSE) the kernel of which is a super propagator. The latter feature leads to the characterization of a composite SC by CPs with multiple binding energies (|W|s). A salient feature of this approach is that it invariably invokes a λ for each of the ion-species that may cause pairing, whence one has the same λs in the equations for any Δ and the corresponding T_c of the SC—as is the case for elemental SCs. Multiple gaps arise in this approach because different combinations of λs operate on different parts of the Fermi surface due to its undulations. Each of the |W|s so obtained is identified with a Δ of the SC. Thus, as shown in [7] , with the input of the values of any two gaps of an SC and a value of its T_c, this approach goes on to shed light on several other values of these parameters. This is not so for MBA, another feature of which is that even when it is employed to deal with the same SC by different authors, the number of bands invoked is not always the same.

The equation for |W₂₀| is:

λ 1 2 a 1 ( θ 1 , μ , W 20 ) + λ 2 2 a 2 ( θ 2 , μ , W 20 ) = [ 3 4 a 3 ( μ , W 20 ) ] 1 / 3 , (2)

where

a 1 ( θ 1 , μ , W 20 ) = R e [ ∫ − k θ 1 k θ 1 d ξ ξ + μ | ξ | + | W 20 | / 2 ] ,   a 2 ( θ 2 , μ , W 20 ) = a 1 ( θ 2 , μ , W 20 ) a 3 ( θ m , μ , W 20 ) = R e [ 4 3 ( μ − k θ m ) 3 / 2 + ∫ − k θ m k θ m d ξ ξ + μ ( 1 − ξ ξ 2 + W 20 2 ) ] . (3)

The equation for |W₂(t)| for 0 < t < 1 is:

λ 1 b 1 ( θ 1 , μ , W 2 ) + λ 2 b 2 ( θ 2 , μ , W 2 ) = [ 3 4 b 3 ( θ m , μ , W 2 ) ] 1 / 3 , (4)

where

b 1 ( θ 1 , μ , W 2 ) = b 11 ( θ 1 , μ , W 2 ) + b 12 ( θ 1 , μ , W 2 ) b 2 ( θ 2 , μ , W 2 ) = b 21 ( θ 2 , μ , W 2 ) + b 22 ( θ 2 , μ , W 2 )

b 1 1 ( θ 1 , μ , W 2 ) = ∫ − k θ 1 0 d ξ μ + ξ tanh [ ( 1 / 2 k t T c ) ( ξ − W 2 ) ] ξ − W 2 b 1 2 ( θ 1 , μ , W 2 ) = ∫ 0 k θ 1 d ξ μ + ξ tanh [ ( 1 / 2 k t T c ) ( ξ + W 2 ) ] ξ + W 2

b 2 1 ( θ 2 , μ , W 2 ) , b 2 2 ( θ 2 , μ , W 2 ) = b 11 ( θ 2 , μ , W 2 ) , b 1 2 ( θ 2 , μ , W 2 )

b 3 ( θ m , μ , W 2 ) = ∫ − μ k θ m μ + ξ [ 1 − ξ tanh [ ( 1 / 2 k t T c ) ξ 2 + W 2 2 ] ξ 2 + W 2 2 ] ,

and the equation for T_c is:

λ 1 2 c 1 ( θ 1 , μ , T c ) + λ 2 2 c 2 ( θ 2 , μ , T c ) = [ 3 4 c 3 ( θ m , μ , T c ) ] 1 / 3 , (5)

where

c 1 ( θ 1 , μ , T c ) = R e [ ∫ − θ 1 / 2 T c θ 1 / 2 T c d x 2 k T c x + μ tanh ( x ) x ] c 2 ( θ 2 , μ , T c ) = c 1 ( θ 2 , μ , T c ) c 3 ( θ m , μ , T c ) = R e [ 2 k T c ∫ − θ m / 2 T c θ m / 2 T c d x 2 k T c x + μ { 1 − tanh ( x ) } ]

In the above equations, θ₁ and θ₂ are the Debye temperatures of the ion-species that cause pairing and are obtained from the Debye temperature θ of the SC as detailed in [7] and [8] ; θ_m is the greater of the temperatures θ₁ and θ₂, and the variation of chemical potential with temperature has been ignored in order to avoid the situation where we have an under-determined set of equations. Thus, we have assumed that the chemical potential μ ( T = 0 ) = μ ( T = T c ) = E F . With the input of |W₂₀|, T_c, and different assumed values of μ = ρ k θ (ρ = 100, 50, 25, 10, 5, etc.), solution of simultaneous Equations (2) and (4) yields, for each value of μ, the corresponding values of the interaction parameters λ₁ and λ₂.

The construct y₀ is defined as

y 0 = ( k θ / P 0 ) 2 m * ( 0 ) / E F , (6)

where m*(0) is the effective mass of superconducting electrons at T = 0 and P₀ their critical momentum.

The exercise carried out above leads to a multitude of values for the set S = {E_F, λ₁, λ₂}, each of which is consistent with the |W₂₀| and T_c values of the SC. In order to find the unique set of values from among them that also leads to the empirical value of j₀ of the SC, we solve the following equation for y₀ [9] for each triplet of {E_F, λ₁, λ₂} values:

1 = λ 1 4 J ( Σ ′ 1 , r 1 y 0 ) + λ 2 4 J ( Σ ′ 2 , r 2 y 0 ) , (7)

where

J ( Σ ′ , y 0 ) = R e [ ∫ 0 1 d x F ( Σ ′ , x , y 0 ) ] , F ( Σ ′ , x , y 0 ) = f 1 ( Σ ′ , x , y 0 ) + f 2 ( Σ ′ , x , y 0 ) f 1 ( Σ ′ , x , y 0 ) = 4 [ − u 1 ( Σ ′ , x , y 0 ) − u 2 ( Σ ′ , x , y 0 ) + u 3 ( Σ ′ , x , y 0 ) + u 4 ( Σ ′ , x , y 0 ) ]

f 2 ( Σ ′ , x , y 0 ) = 2 ln [ 1 + u 1 ( Σ ′ , x , y 0 ) 1 − u 1 ( Σ ′ , x , y 0 ) 1 + u 2 ( Σ ′ , x , y 0 ) 1 − u 1 ( Σ ′ , x , y 0 ) 1 − u 3 ( Σ ′ , x , y 0 ) 1 + u 3 ( Σ ′ , x , y 0 ) 1 − u 4 ( Σ ′ , x , y 0 ) 1 + u 4 ( Σ ′ , x , y 0 ) ] u 1 ( Σ ′ , x , y 0 ) = 1 − Σ ′ x / y 0 , u 2 ( Σ ′ , x , y 0 ) = 1 + Σ ′ x / y 0 u 3 ( Σ ′ , x , y 0 ) = 1 − Σ ′ ( 1 − x / y 0 ) , u 4 ( Σ ′ , x , y 0 ) = 1 + Σ ′ ( 1 − x / y 0 ) Σ ′ = k θ / E F , Σ ′ i = k θ i / E F , r i = θ i / θ .

The operator Re ensures that the integrals yield real values even when expressions under the radical signs are negative (as happens for the heavy fermion SCs).

Corresponding to each value of y₀ obtained by solving (6) with the input of {θ, θ₁, θ₂, E_F, λ₁, λ₂}, we can calculate several superconducting parameters in terms of θ, E_F, y₀, the gram-atomic volume v_g and the electronic heat constant/Sommerfeld coefficient γ₀ at T = 0 by employing the following relations, see [8] and for a correction, [10] :

s ( 0 ) ≡ m * ( 0 ) / m e = A 1 ( γ 0 / v g ) 2 / 3 E F 1 / 3 (8)

N s 0 ( E F ) = A 2 ( γ 0 / v g ) E F (9)

P 0 ( E F ) = A 3 θ y 0 ( γ 0 / v g ) 1 / 3 E F 2 / 3 (10)

V c 0 ( E F ) = A 4 θ y 0 1 ( γ 0 / v g ) 1 / 3 E F 1 / 3 (11)

j 0 ( E F ) = N s 0 ( E F ) 2 e * V c 0 ( E F ) = A 5 θ y 0 ( γ 0 / v g ) 2 / 3 E F 2 / 3 , ( e * = 2 e ) (12)

where m_e is the free electron mass and e the electronic charge,

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   cm ⋅ K − 1 / 3 ,   A 4 ≅ 1.406 × 10 8   eV 2 / 3 ⋅ sec − 1 ⋅ K − 5 / 3 ,

A 5 ≅ 6.146 × 10 − 4   C ⋅ eV − 4 / 3 ⋅ K 1 / 3 ⋅ s − 1 ,

and N_s0 is the number density of CPs and V_c0 their critical velocity at T = 0. Comparison of the j₀-values so obtained with the experimental value of j₀ then leads to the desired unique set of {E_F, λ₁, λ₂}-values that is consistent with the empirical values of |W₂₀|, T_c, and j₀ of the SC.

When t ≠ 0, the equation for y(t) is obtained from [9]

1 = λ 1 4 ∫ 0 1 d x [ I 1 ( x , θ 1 , E 2 ) + I 2 ( x , θ 1 , E 2 ) ]           + λ 2 4 ∫ 0 1 d x [ I 1 ( x , θ 2 , E 2 ) + I 2 ( x , θ 2 , E 2 ) ] , (13)

where

I 1 ( x , θ , E 2 ) = ∫ − E 1 + E 2 x E 1 − E 2 x d ξ     ϕ ( ξ ) tanh [ ( β / 2 ) ( ξ + E 2 x ) ] I 2 ( x , θ , E 2 ) = ∫ − E 1 + E 2 x E 1 − E 2 x d ξ     ϕ ( ξ ) tanh [ ( β / 2 ) ( ξ − E 2 x ) ] ϕ ( ξ ) = 1 + ξ / E F / ( ξ + E 3 ) , E 1 = k θ E 2 ( t ) = P c ( t ) E F / 2 m * ( t ) , E 3 ( t ) = P c 2 ( t ) / 8 m * ( t ) , β = 1 / k t T c .

In terms of y ( t ) = E 1 / E 2 ( t ) = [ k θ / P c ( t ) ] 2 m * ( t ) / E F , we have (12) as:

1 = λ 1 4 ∫ 0 1 d x   f ( x , θ 1 , E F , r 1 y ) + λ 2 4 ∫ 0 1 d x   f ( x , θ 2 , E F , r 2 y ) , (14)

where

f ( x , θ , E F , y ) = ∫ − E 1 ( 1 − x / y ) E 1 ( 1 − x / y ) d ξ     ϕ ( ξ , E F , y ) { tanh [ θ 2 t T c y ( ξ y k θ + x ) ] + tanh [ θ 2 t T c y ( ξ y k θ − x ) ] } E 1 = k θ , ϕ ( ξ , E F , y ) = 1 + ξ / E F / [ ξ + E 3 ( y ) ] , E 3 ( y ) = E 1 2 / 8 m * α 2 y 2 , r i = θ i / θ .

Each value of y(t) obtained by solving (13) for different values of the set S = {E_F, λ₁, λ₂} leads to the corresponding value of j_c(t) via (11), and to the values of the associated parameters via (7)-(10).

Reported in [1] are various superconducting properties for six samples of aluminium strips at T = 0. Among these, T_c and j₀ are determined experimentally, whereas values of some other parameters such as coherence length ξ and the London penetration depth λ_L are model-dependent derived properties.

In order to show how the GBCSEs-based approach works in such a situation, we give below the sequential steps that are followed for Sample 1 in [1].

1) The T_c of the sample is 1.196 K. We take its Debye temperature θ to be 428 K [11]. Employing these values and putting λ₂ = 0 (as is appropriate for a 1 PEM), solution of (4) for some typical values of E_F are: λ₁ = 0.1665 for any value of E_F = 10 - 100kθ; λ₁ = 0.1666 for E_F = 5kθ; λ₁ = 0.1670 for E_F = 2kθ.

2) For some select pairs of {E_F, λ₁} values obtained above, we now solve (6) for the corresponding value of y₀ and obtain the following results (in the parentheses are given the {E_F, λ₁} values): y₀ = 149.8 {50kθ, 0.1665}; y₀ = 149.9 {25kθ, 0.1665}; y₀ = 149.9 {10kθ, 0.1665}; y₀ = 149.6 {5kθ, 0.1666}; y₀ = 149.5 {2kθ, 0.1670}.

3) For each of the above pairs of {E_F, y₀} values, we now calculate j₀ via (11). For this purpose, besides the value of θ, we require the values of γ₀ and v_g which are taken as: γ₀ = 1.36 mJ/mol-K² [11] , v_g = 10 cm³/gram-atom. The results in A/cm² with the corresponding {E_F, y₀} values given in parentheses are: 2.37 × 10⁷ {50kθ, 149.8}, 1.49 × 10⁷ {25kθ, 149.8} 8.09 × 10⁶ {10kθ, 149.9}, 5.11 × 10⁶ {5kθ, 149.6}, 2.77 × 10⁶ {2kθ, 149.5}.

4) Among the above values, j₀ = 1.49 × 10⁷ A/cm² corresponding to {E_F = 25kθ, y₀ = 149.8} is closest to the experimental value of 1.53 × 10⁷ A/cm². By fine-tuning the value of E_F and repeating the above exercise, we find that the experimental value of j₀ is obtained exactly when E_F = 26kθ = 0.959 eV and y₀ = 149.8.

5) With E_F fixed at this value, we can find W₁₀ via (2) (with λ₂ = 0).

6) With E_F and y₀ of the sample fixed as above, we can calculate the values of s(0), N_s(0), and V_c0 via (7), (8) and (10), respectively, and the Fermi velocity V_F0 at T = 0 via the relation

( 1 / 2 ) m * ( 0 ) V F 0 2 = E F . (15)

Besides these parameters, we can now also calculate the coherence length ξ₀ at T = 0 via

ξ 0 = ℏ V F 0 / π | W 10 | , (16)

and the London penetration depth λ_L0 at T = 0 via

λ L 0 = m * ( 0 ) c m 3 ( ℏ c ) 2 / 4 π n s ( 0 ) e 2 , (17)

where units for m* and n_s are electron-volt and cm⁻³, respectively, e = (137.03604)^−1/2, cm = 5.067728861 × 10⁴(ħc) eV⁻¹ and, for convenience of the reader, we have inserted the requisite factors of ħ and c in order to obtain the value of λ_L0 in cgs units. The results of these calculations are given in Table 1, which also gives similar results for the remaining five samples dealt with in [1].

Insofar as j_c(t) is concerned, Romijn et al. [1] have given results for one of their samples (Sample 5) in their Figure 4, which is a plot of their experimental values of [ j c ( t ) / j c ( 0 ) ] 2 / 3 along with their counterparts as obtained via the phenomenological Bardeen Equation (1) and the KL theory [3]. It follows from this

graph that the absolute experimental values of j_c(t) are given fairly accurately by

j c ( t ) | exp ≃ j 0 | exp ( 1 − t 2 ) 3 / 2 , (18)

where j 0 | exp = 1.07 × 10 7 A / cm 2 .

We recall that j₀ of Sample 5 in our approach was calculated via the following values of the associated parameters: θ = 428 K, T_c = 1.356 K, λ₁ = 0.1701, E_F = 12.6kθ, y₀ = 132.0, γ₀ = 1.36 mJ/mol K⁻², and v_g = 10 cm³/gram-atom. In order now to calculate j_c(t) for this sample, we need to take into account the T-dependence of all these parameters. It seems reasonable to assume that among them, θ, E_F and v_g retain the values employed for them at T = 0. This assumption enables us to calculate y(t) for any “t” via (13) (with λ₂ = 0); the resulting values are given in Table 2. We could now calculate j_c(t) if we knew γ(t), but about which we have no information. However, we know that the heat capacity of an SC has a marked non-linear dependence on t as discussed, e.g. in general in [12] and for superconducting Ga in ( [13] , p. 411). We are hence led to calculate γ(t) with the input of j_c(t)—rather than the other way around—via the following equation

j c ( t ) | exp = j c ( θ , y ( t ) , γ ( t ) , v g , E F ) | Th , (19)

the LHS of which for any “t” is taken to be given by (17) and the RHS is calculated via (11) with the input of y(t) obtained by solving (13). These values of γ(T) are included in Table 2. Considered together with the values of y(t), they will be shown below to provide a microscopic justification of Bardeen’s phenomenological Equation (1). It is also remarkable that y(t) and γ(t) enable one to obtain quantitative estimates of several other t-dependent superconducting parameters, viz., s(t) = m*(t)/m_e, n_s(t) = N_s(t)/N_s0, v_F(t) = V_F(t)/V_F0, v_c(t) = V_c(t)/V₀, ξ_r(t) = ξ(t)/ξ₀, and λ_Lr(t) = λ_L(t)/λ_L0. The plots of these are discussed below.