The PE is obtained via a 4-d Bethe-Salpeter equation (BSE) the 4^th dimension of which is employed to temperature-generalize it via the Matsubara prescription. Subjecting the resulting 3-d equation to the Landau quantization scheme further generalizes it to include an applied magnetic field leading to the quantization of the transverse components of momentum into Landau levels. Finally, the PE employed here is obtained from this 1-d equation by putting the binding energy/gap Δ = 0, whence it gives the value of the critical magnetic field H_c at any T or, equivalently, T_c corresponding to any H. Derived in [7] and subjected to a correction [8] , this equation is reproduced below from the latter reference where it was shown to provide the basis of a new microscopic approach for dealing with the T- and H-dependent critical current densities of SCs.

$1-{\lambda }_m{\displaystyle \underset{L_1\left(\mathrm{..}\right)}{\overset{L_2\left(\mathrm{..}\right)}{\int }}\frac{\text{d}\xi }{\sqrt{1+\xi /\mu }}{\displaystyle \underset{n=0}{\overset{N_{L1}\left(\mathrm{..}\right)}{\sum }}\frac{\tanh \left[\frac{\xi }{2kT}+\frac{\hslash \Omega \left(H_c\right)}{2kT}\left(n+1/2\right)\right]}{\xi +\hslash \Omega \left(H_c\right)\left(n+1/2\right)}}}=0,$ (3)

where

$\begin{array}{l}{\lambda }_m=\frac{e{H}_{c}V}{16{\pi }^2}\sqrt{\frac{2\eta m_e}{\mu }},\mu =\rho k\theta ,L_1\left(\mathrm{..}\right)=\frac{-k\theta }{3}\left(2\rho +1\right),L_2\left(\mathrm{..}\right)=\frac{k\theta }{3}\left(-2\rho +1\right)\\ N_{L1}\left(\mathrm{..}\right)=floor\left[\frac{2}{3}\frac{\left(\rho +1\right)k\theta }{\hslash \Omega \left(H_c\right)}-\frac{1}{2}\right],\Omega \left(H_c\right)={\Omega }_0{H}_c/\eta ,{\Omega }_0=1.7588{\text{s}}^{-1}\cdot {\text{G}}^{-1}\end{array}$

The parameter λ_m is the magnetic interaction parameter corresponding to temperature T, applied field H and the chemical potential μ which has been parametrized in terms of the Debye temperature of the SC as ρkθ, where k is the Boltzmann constant and ρ a free parameter; - V (V > 0) is the usual BCS model interaction due to the ion-lattice and Coulomb repulsion between electrons and finally, m_e is the mass of an electron and η its enhancement factor.

Also reproduced from [8] , the NE is:

$N_s\left(\mathrm{...}\right)=C_1{\left(h{H}_c\right)}^{3/2}{\displaystyle \underset{0}{\overset{\sqrt{L\left(\mathrm{..}\right)}}{\int }}{\displaystyle \underset{n=0}{\overset{N_2\left(\mathrm{..}\right)}{\sum }}F\left(\mathrm{...}\right)\text{d}z,}}\left(h\ne 0\right)$ (4)

where

$\begin{array}{l}{C}_1=2.1213\times {\text{10}}^9,t=T/T_c,h=H/H_c,L\left(\mathrm{..}\right)=\frac{1}{3}\frac{\left(\rho q+1\right)k\theta }{\hslash \Omega \left(h_c\right)}\\ N_2\left(\mathrm{..}\right)=floor\left[\frac{2}{3}\frac{\left(\rho q+1\right)k\theta }{\hslash \Omega \left(h_c\right)}-\frac{1}{2}\right]\end{array}$

$F\left(\mathrm{...}\right)=\left[1-\tanh \left\{\frac{\hslash \Omega \left(h\right)}{2kt{T}_c}\left(n+1/2+z^2-\frac{q\rho k\theta }{\hslash \Omega \left(h\right)}\right)\right\}\right]$

The PE and NE are supplemented by the equation for the London penetration depth

${\lambda }_L\left(0\right)=\sqrt{\frac{{\epsilon }_0\eta m_e{c}^2}{N_s{e}^2}},$ (SI Units) (5)

where ε₀ = 8.85 × 10⁻¹² F∙m⁻¹, c is the velocity of light, e the electronic charge, and N_s is as given by (4).

In order to show that the properties of an SC in an external field are determined predominantly by $\mu \left(t_1\right)$, we proceed as follows.

1) Solve (3) for any SC to obtain the value of λ_m1 with the input of its empirically listed values of T_c, H_c, an assumed value of μ₁ = ρkθ (ρ a free parameter), t₁ = 0.9 and the relation

$h_c\left(t\right)=H_c\left(t\right)/H_c=1-t^2,$ (6)

which for the most part has been found to be sound empirically and leads to h_c1 = 0.19. Employment of this relation has been necessitated because, in general, the value of h_c1 at t₁ is not known. Note that the choice of t₁ = 0.9 does away with the need to specify the self-field, which would be needed for t = 1 because then h_c1 = 0, whereas a non-zero value of the latter is required for the employment of (4).

2) After λ_m1 is determined as noted in 1), solve (3) to determine the value of μ₀ corresponding to the values of t₀ = 0.1 and h_c0 = 0.99 via (6). The reason for choosing t = 0.1 and not 0 is that the listed value of any parameter at t = 0 is invariably one that is extrapolated from a low value of t because no experiment has ever been performed exactly at T = 0. This will be further discussed below. Since μ₁ was parametrized as ρkθ, if we define μ₀ = qμ₁, then it means that we are now solving the equation for q by replacing ρ by qρ in the earlier equation that yielded the value of λ_m1. Of course, we also need the value of λ_m0 in order to obtain the value of q, which is easily obtained from the expression for λ_m given below (3) by noting that

$\frac{{\lambda }_{m0}}{{\lambda }_{m1}}=\frac{h_{c0}}{h_{c1}}\frac{1}{\sqrt{q}},$

since ${\mu }_0=q{\mu }_1$. For the sake of concreteness, we write below the equation from which q is calculated. In terms of the dimensionless variable $x=\xi /2kt{T}_c$, this equation is

$1-{\lambda }_{m1}\frac{h_{c0}}{h_{c1}}\frac{1}{\sqrt{q}}{\displaystyle \underset{x_1\left(\mathrm{..}\right)}{\overset{x_2\left(\mathrm{..}\right)}{\int }}{\displaystyle \underset{n=0}{\overset{N_{L0}\left(\mathrm{..}\right)}{\sum }}\frac{\text{d}x}{\sqrt{1+\frac{2t_0{T}_c}{q\rho \theta }}x}\frac{\tanh \left[x+\left(n+1/2\right)\frac{\hslash \Omega \left(h_{c0}\right)}{2k{t}_0{T}_c}\right]}{\left[x+\left(n+1/2\right)\frac{\hslash \Omega \left(h_{c0}\right)}{2k{t}_0{T}_c}\right]}}}=0,$ (7)

where

$\begin{array}{l}{x}_1\left(\mathrm{..}\right)=\frac{-\theta }{6t_0{T}_c}\left(2\rho q+1\right),x_2\left(\mathrm{..}\right)=\frac{\theta }{6t_0{T}_c}\left(-2\rho q+1\right)\\ N_{L0}\left(\mathrm{..}\right)=floor\left[\frac{2}{3}\frac{k\theta }{\hslash \Omega \left(h_{c0}\right)}\left(\rho q+1\right)-\frac{1}{2}\right].\end{array}$

3) For the remainder of our procedure, it is convenient to deal with a specific SC, say Nb, for which λ_L(0) = 52 nm [1] . With the values of θ, η, T_c and H_c given in Table 1 , the choice of the free parameter ρ = 10 leads via (3) to λ_m1 = 1.212 × 10⁻⁶ and thence to q = 2.607, N_s = 2.81 × 10²⁸ m⁻³ and λ_L(0) = 109.8 nm via (4) and (5), respectively. Since the last of these values is more than twice its listed value, we now repeat the steps carried out for ρ = 10 by progressively increasing ρ. Thus, we find that ρ = 30.7 leads to λ_L(0) = 51.96 nm. The values of the other parameters corresponding to these are given in Table 1 , which includes the final results for not only Sn and Pb, but also MgB₂ and YBCO.

Continued

The procedural details for dealing with these SCs are similar to those for the elemental SCs, except that in the BSE-based approach which leads to a generalization of the BCS equations (GBCSEs) via the employment of a superpropagator, explanation of the values of its T_c and multiple gaps requires invoking more than a 1-phonon exchange mechanism (1PEM) for the formation of Cooper pairs. This is in contrast to the situation for elemental SCs for which 1PEM suffices. Hence, the question is: Do we require, say, 2PEM to deal with the properties of a composite SC in a magnetic field? The answer to this question is: not necessarily, because for the SCs considered here, the magnetic interaction parameter, i.e., λ_m in (3), due to the B-ions in MgB₂ (Mg plays a secondary role via the proximity effect), or due to the Y- or Ba-ions ions in YBCO, turns out to be considerably smaller (and well below the Bogoliubov upper limit of 0.5) than any of the non-magnetic interaction parameters that occur in the equations for their T_c and the gaps [11] .

For MgB₂, we now need to resolve θ(MgB₂) = 815 K which is the mean of its values given in [12] into θ_B and θ_Mg. The basic idea here is due to Born and Karmann [13] [14] who had pointed out a long time ago that elastic waves in an anisotropic solid travel with different velocities in different directions and are hence characterized by different Debye frequencies or temperatures. Employing the double-pendulum model for the resolution of θ(MgB₂) into the θs of its constituents [15] , we obtain [11] θ_B = 1062 K and θ_Mg =322 K (which we will not need). The basis for the value of each of the other properties of MgB₂ noted in column 1 of Table 1 is as follows

$T_c=39\text{K}$ [12] , $\eta =\text{3}\text{.1}$ [16]

$H_{c2}=2.5\times {10}^4\text{G}\left(H\left|\right|c\right)$ [12] , ${\lambda }_L^{ab}\left(0\right)=140\text{nm}$ [17] (8)

When the 1PEM is considered to be due to the Y ions in YBCO, the values of the above parameters are:

θ_Y = θ(YBCO) = 410 K [1] (because the sub-lattice that contains the Y ions has no other constituent)

$T_c=92\text{K}$ [1] , $\eta =3.0$ [18]

$H_{c2}=112\times {10}^4\text{G}\left(H\left|\right|c\right)$ [19] , ${\lambda }_L^{ab}\left(0\right)=89\text{nm}$ [1] (9)

It is seen from Table 1 that when the 1PEM is considered to be due to the Y ions, λ_L(t₀) ≈ 89 nm is obtained when μ₁ = 0.240 eV and that nearly the same value for λ_L(t₀) results when the IPEM is considered to be due to the Ba ions with μ₁ = 0.255 eV.

Of the samples the properties of which were noted in (2), we have already dealt in detail with Sample 2 that has T_c = 92 K and ${\lambda }_L^{ab}\left(0\right)=89\text{nm}$. We now deal with Sample 1 of the same SC which is characterized by T_c = 66 K and ${\lambda }_L^{ab}\left(0\right)=260\text{nm}$. H_c2 of this sample may be estimated to be 91 × 10⁴ G, since H_c2 = 87 × 10⁴ G for the sample for which T_c = 62 K - vide (1). The last row in Table 1 gives the values of all the parameters corresponding to ${\lambda }_L^{ab}\left(0\right)$ of Sample 1.