On the Role of Chemical Potential in Determining the Temperature Dependent Critical Magnetic Field and the Penetration Depth of Superconductors

Gulshan Prakash Malik^1,2, Vijaya Shankar Varma^1,3

¹Gurugram, India.
²Theory Group, School of Environmental Sciences, Jawaharlal Nehru University, New Delhi, India.
³Department of Physics and Astrophysics, University of Delhi, Delhi, India.

DOI: 10.4236/wjcmp.2024.144009                

Abstract

Dealing with both elemental and high-T_c superconductors (SCs) - Sn, Nb and Pb belonging to the former category, and MgB₂ and different samples of YBCO to the latter - we show that the difference in the values of their critical magnetic field H_c1,c2 and the penetration depth λ_L(0) is, remarkably, attributable predominantly to the difference in the values of a single parameter, viz., the chemical potential (μ) close to their critical temperatures (T_cs). Based directly on the dynamics of pairing in a magnetic field and the corresponding number equation, our approach relates H_c1,c2 of an SC with the following set of its properties: S₁ = {μ, T_c, Debye temperature, effective mass of the electron, magnetic interaction parameter, Landau index}. Hence, it provides an alternative to the approach followed by Talantsev [Mod. Phys. Lett. B 33, 1950195 (2019)] who has shown by ingeniously combining the results of various well-established theories that H_c2 of an SC can be calculated via four different equations, each of which invokes two or more properties from its sample-specific set S₂ = {T_c, gap, coherence length, λ_L(0), jump in sp. ht.}, which is radically different from S₁.

Keywords

Pairing and Number Equations Incorporating Temperature, Chemical Potential and Magnetic Field, Critical Magnetic Field, Penetration Depth, Sn, Nb, Pb, MgB₂, YBCO

1. Introduction

It is well known that the properties of a superconductor (SC) differ not only from one SC to another, but also from sample to sample of the same SC - depending on the sample geometry, size, nature of doping, manner of preparation, etc. Thus, for example, two samples of YBa₂Cu₃O₇ have been found to have the values of their critical temperature (T_c) and the upper critical magnetic field $H_{c2}^c\left(0\right)$ at temperature T = 0 when the applied field H is along the c-direction as given below [1].

YBa₂Cu₃O_6.5: T_c = 62 K, $H_{c2}^c\left(0\right)=87\times {\text{10}}^4\text{G}$

YBa₂Cu₃O_7-δ (with δ ≈ near-optimal value of doping): T_c = 92.4 K,

$H_{c2}^c\left(0\right)=34\times {\text{10}}^4\text{G}$ (1)

Pertaining to a different pair of samples (with unspecified values of δ) of the same SC and direction of the applied field, the values of T_c and the London penetration depth ${\lambda }_L^{ab}\left(0\right)$ below provide another example [1].

Sample 1: $T_c=66\text{K}$ , ${\lambda }_L^{ab}\left(0\right)=260\text{ nm}$

Sample 2: $T_c=92\text{K}$ , ${\lambda }_L^{ab}\left(0\right)=89\text{ nm}$ (2)

Employing the examples of both elemental SCs (Sn, Nb and Pb) and high-T_c SCs (MgB₂ and two samples of YBCO), it is the purpose of this paper to show that the difference in the above-noted properties of these SCs can be understood predominantly on the basis of the difference in the values of a single parameter that characterizes them. This parameter, remarkably, is μ₁, i.e., the value of the chemical potential μ at t₁ ($\equiv$ T₁/T_c) close to 1 (say, 0.9) and hence amenable to experimental verification. The ramifications/implications of this result will be discussed. Before we do so, however, we give below a brief account of some of the earlier approaches to calculating H_c2 of an SC in order to highlight the difference between them and the approach followed in this paper.

The most widely employed approach for the calculation of H_c2 (B_c2 in the SI system of units) is due to Werthamer, Helfand and Hohenberg (WHH) [2], which is based on the electron spin and spin-orbit effects. The WHH equation is

$\ln \left(\frac{T}{T_c\left(B=0\right)}\right)=\psi \left(\frac{1}{2}\right)-\psi \left(\frac{1}{2}+\frac{\hslash D{B}_{c2}\left(T\right)}{2{\varphi }_{0}kT}\right),$

where ψ is the Euler’s digamma function, D the diffusion constant for the normal electrons/holes in the conduction band, ${\varphi }_0=2.07\times 10^{-15}\text{Wb}$ the flux quantum and k the Boltzmann constant. An older equation for B_c2 in the classical two-fluid Gorter-Casimir model [3] is

$B_{c2}\left(t\right)=\frac{{\varphi }_0}{2\pi {\xi }_0^2}\left(1-t^2\right),$

where ξ₀ is the coherence length at t = 0. A variant of the above equation is the Jones-Hulm-Chandrasekhar model [4]

$B_{c2}\left(t\right)=\frac{{\varphi }_0}{2\pi {\xi }_0^2}\frac{1-t^2}{1+t^2}.$

Finally, the following equation for B_c2 given by Gor’kov [5] is also said to provide a good fit to the data over the entire range of temperature

$B_{c2}\left(t\right)=B_c\left(t\right)\frac{\sqrt{2}}{1.77}\frac{{\lambda }_0}{{\xi }_0}\left(1.77-0.53t^2+0.07t^4\right),$

where $B_c\left(t\right)$ is the thermodynamic critical field.

It is interesting to note that combining each of the models noted above with a mix of the other properties of the SC, e.g., its gap Δ and the jump in the specific heat, Talantsev [6] has recently derived four new equations for B_c2 each of which leads to nearly as good a fit to the empirical data of compressed H₃S as the other. It is notable that all of these approaches including that of Talantsev determine B_c2 indirectly via the effects it has on one or more properties of the SC such as ξ₀ and λ₀. In contrast with these, the approach followed in this paper determines H_c2 directly via the dynamics of pairing, i.e., by employing an H_c2-dependent pairing equation (PE) together with the corresponding number equation (NE). Below, we will return to a further discussion of Talantsev’s work and draw attention to how our approach complements his approach.

The PE and the NE on which the present study is based are given in the next section. Both of these equations are μ-, T- and H-dependent. These are supplemented by the equation for λ_L(0). Procedural details of the application of these equations to both elemental and high-T_c SCs are given in Section 3. The final section sums up our findings.

2. The Core Equations

2.1. The μ-, T- and H-Dependent Pairing Equation (PE)

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.

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

2.2. The μ-, T- and H-Dependent Number Equation (NE)

Also reproduced from [8], the NE is:

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

3. Procedural Details

In the following

$\begin{array}{l}{t}_1=T_1/T_c,h_{c1,c2}\left(t_1\right)=H_{c1,c2}\left(T_1\right)/H_{c1,c2}\left(0\right)\\ t_0=T_0/T_c,h_{c1,c2}\left(t_0\right)=H_{c1,c2}\left(T_0\right)/H_{c1,c2}\left(0\right),\end{array}$

where t₁ = 0.9 and t₀ = 0.1 for reasons to be specified.

3.1. Elemental SCs

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

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

Table 1. The empirical values for elemental SCs in column 1 are taken from Poole [1] and the values of η from Kittel [9], except for Nb for which it is taken from Ashcroft and Mermin [10]. The equations employed to obtain the values of various parameters are as follows: λ_m1 and N_L1 via (3); q, λ_m0(t₀) and N_L0 via (7) and N_s and λ_L(t₀) via (4) and (5), respectively. For the values of the parameters pertaining to MgB₂ and YBCO in column 1, see (8) and (9), respectively. YBCO/Y and YBCO/Ba mean that the 1PEM is due to the Y- and Ba-ions, respectively.

SC	$\rho$	${\mu }_1=\rho k\theta$ (eV)	${\lambda }_{m1}$	$N_{L1}$	$q=\frac{{\mu }_0}{{\mu }_1}$	${\lambda }_{m0}$	$N_{L0}$	$N_s\left(t_0\right)\times {10}^{28}$ (m−3)	${\lambda }_L\left(t_0\right)$ (nm)
θ, Tc (K)
$\eta =m_e^{\ast }/m_e$
Hc1(G)
λL(0) (nm)
Sn
195, 3.72	148	2.49	2.192 × 10−6	3,134,994	1.975	8.125 × 10−6	1,184,352	2.029	41.87
1.26
305
42
Pb
96, 7.2	249	2.06	9.075 × 10−6	1,543,592	2.27	3.139 × 10−5	670,971	3.675	38.9
1.97
800
39
Nb
276, 9.25	30.7	0.730	1.074 × 10−6	1,735,552	2.356	3.645 × 10−6	770,501	12.55	51.96
12
1580
52
MgB2
1062, 39	2.32	0.212	2.331 × 10−5	11,418	3.055	6.948 × 10−5	5538	0.447	139.9
3.1
2.5 × 104
140
YBCO/Y
410, 92	6.77	0.239	4.427 × 10−3	222	5.506	9.831 × 10−3	210	1.070	88.89
3.0
112 × 104
89

Continued

YBCO/Ba
478 [8], 92	6.2	0.255	3.630 × 10−3	240	5.139	8.343 × 10−3	210	1.072	88.88
3.0
112 × 104
89
YBCO/Y
410, 66	1.264	0.045	4.245 × 10−3	79	6.516	8.666 × 10−6	62	0.125	260.0
3.0
91 × 104
260

3.2. MgB₂ and YBCO

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.

3.3. Dealing With Two Samples of YBCO that Have Significantly Different Values of T_c and λ_L(0)

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.

4. Summing Up

It has been shown in this paper that one can choose for any SC, via the choice of ρ, any value of μ(t₁) from among a multitude of them and find a value of μ(t₀) corresponding to it which is consistent with the empirical value of H_c1,c2(t₀) of the SC for t₀ close to 0. However, demanding that the so-obtained value of H_c1,c2(t₀) should also lead to the empirical value of λ_L(t₀) of the SC fixes μ₁ uniquely.

It is seen from Table 1 that the value of μ₀ = qμ₁ (which is ≈ Fermi energy, E_F) for MgB₂, or any sample of YBCO, is considerably smaller than its value for the elemental SCs. This is in accord with the general belief that high-T_c superconductivity is governed by low values of E_F; see, e.g., [20].

It is interesting to note that we have adopted in this paper a variant of a part of the template given by Dogan and Cohen [21] to deal with various properties of SCs where the route to the value of λ_L(0) is provided by the following equations

${\xi }_0=\sqrt{\frac{{\varphi }_0}{2\pi H_{c2}\left(0\right)}}$ (10)

$v_F=\frac{\pi {\Delta }_0{\xi }_0}{\hslash }$ (11)

$N_s\left(0\right)=\frac{1}{3{\pi }^2}{\left(\frac{m_e^2{v}_F^2}{{\hslash }^2}\right)}^{3/2},$ (12)

followed by (5) for the calculation of λ_L(0); ${\varphi }_0$ above denotes the magnetic flux quantum and Δ₀ and ξ₀ denote the gap and the coherence length, respectively. The application of the above equations to YBCO with H_c2(0) = 112 T, Δ₀ = 20 meV leads to

${\xi }_0=1.71\text{nm, }{v}_F=1.64\times {10}^5\text{m}/\text{s}\text{, }{N}_s=9.54\times {10}^{25}{\text{m}}^{-3},{\lambda }_L\left(0\right)=544\text{nm}$ (13)

In the approach followed here on the other hand, m_e is replaced by ηm_e and the values of ξ₀ and Δ₀ are not required. Solution of (3) with the input of μ₁ then leads to the values of λ_m0 and N_L0; that of (7) to the values of q, λ_m0 and N_L0; N_s(t₀) is obtained via (4) which is employed in lieu of (12) and, finally, λ_L(t₀) is calculated via (5). It is notable that our approach leads to v_F(YBCO) = 3.9 × 10⁵ m/s via μ₀ = qμ₁ and $v_F=\sqrt{2{\mu }_0/\eta m_e}c$ (μ₀ and m_e in units of electron-Volt) which is very close to the upper limit of the range (2.5 × 10⁵ – 3.8 × 10⁵ m/s) of the universal Fermi velocity of hydrides [22].

Since the lowest value of t adopted in this paper is 0.1, it is tacitly assumed that the values of the parameters on which λ_L depends change insignificantly on extrapolation from t = 0.1 to t = 0. This flattening of the H_c2(t) vs. t plot for such values of t is in accord with the results obtained by Talantsev [6] for compressed H₃S by employing four different equations based on well-established theories. Each of these equations invoked two or more parameters from the sample-specific set of the SC comprising {T_c, gap, coherence length, λ_L(0) and jump in sp. ht.} However, departures from this type of behavior, i.e., a steep rise in the value of H_c2(t) as t → 0 which is reminiscent of the divergence of ξ as t → 1, seem to have been empirically observed recently. For this reason, we believe that it is a good practice while quoting the value of any parameter to specify the lowest temperature at which it has been empirically determined (rather than reporting its extrapolated value) as has been done by Minkov, et al. [23] who have quoted the value of T as 10 K while reporting for compressed H₃S the values of ξ₀, λ_L, etc. For a further discussion of the possible departure of the value of H_c2(t) from its value from, say t = 0.1, to lower values of t, we draw attention to [24] where a study of the H_c2(t) of compressed H₃S and the issue of whether or not it satisfies the criterion of the Meissner effect was carried out by following the same approach as in this paper.

A significant result of this paper is that if we could find empirically for any SC the values of h_c1/h_c2 at t₁ and the unique value of μ(t₁) that leads via μ(t₀) to the value of its λ_L(t₀) by employing the equations that have been given, then we shall have a new, dynamics-based approach to calculate H_c2 as an alternative to the existing approaches. We believe that this result is empirically verifiable. It is interesting to draw attention to yet another approach for the calculation of H_c2 which is based on the thermodynamics of an SC in a magnetic field [25] and hence differs from all the other approaches including the one presented in this paper. It is worth noting that these different approaches ought to be viewed as complementing each other - rather than as competing approaches - because they shed light on different features of the same phenomenon. Finally, in so far as the practical applications are concerned, the study reported here suggests that by controlling the value of μ – which is a problem that belongs to the realm of chemical engineering - we could fabricate SCs that have bespoke properties.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Poole Jr., C.P. (2000) Handbook of Superconductivity. 2nd Edition, Academic Press.
[2]	Werthamer, N.R., Helfand, E. and Hohenberg, P.C. (1966) Temperature and Purity Dependence of the Superconducting Critical Field, Hc2. III. Electron Spin and Spin-Orbit Effects. Physical Review, 147, 295-302. https://doi.org/10.1103/physrev.147.295
[3]	Gorter, C.J. and Casimir, H. (1934) On Supraconductivity I. Physica, 1, 306-320. https://doi.org/10.1016/s0031-8914(34)90037-9
[4]	Jones, C.K., Hulm, J.K. and Chandrasekhar, B.S. (1964) Upper Critical Field of Solid Solution Alloys of the Transition Elements. Reviews of Modern Physics, 36, 74-76. https://doi.org/10.1103/revmodphys.36.74
[5]	Gor’kov, L.P. (1960) The Critical Superconducting Field in the Superconductivity Theory. Soviet Physics JETP, 10, 593-599.
[6]	Talantsev, E.F. (2019) Classifying Superconductivity in Compressed H3S. Modern Physics Letters B, 33, Article ID: 1950195. https://doi.org/10.1142/s0217984919501951
[7]	Malik, G.P. (2010) On Landau Quantization of Cooper Pairs in a Heat Bath. Physica B: Condensed Matter, 405, 3475-3481. https://doi.org/10.1016/j.physb.2010.05.026
[8]	Malik, G.P. and Varma, V.S. (2021) A New Microscopic Approach to Deal with the Temperature-and Applied Magnetic Field-Dependent Critical Current Densities of Superconductors. Journal of Superconductivity and Novel Magnetism, 34, 1551-1561. https://doi.org/10.1007/s10948-021-05852-8
[9]	Kittel, C. (1974) Introduction to Solid State Physics. 4th Edition, Wiley Eastern.
[10]	Ashcroft, N.W. and Mermin, N.D. (1976) Solid State Physics. Cengage Learning.
[11]	Malik, G.P. (2010) Generalized BCS Equations: Applications. International Journal of Modern Physics B, 24, 3701-3712. https://doi.org/10.1142/s0217979210055858
[12]	Buzea, C. and Yamashita, T. (2001) Review of the Superconducting Properties of Mgb2. Superconductor Science and Technology, 14, R115-R146. https://doi.org/10.1088/0953-2048/14/11/201
[13]	Born, M. and Von Karmann, T. (1912) Vibrations in Space Gratings (Molecular Frequencies). Zeitschrift für Physik, 13, 297-309.
[14]	Born, M. and Von Karmann, T. (1913) ZurTheorie der Spezifis chen Waermen. Zeitschrift für Physik, 14, 15.
[15]	Malik, G.P. (2022) The Debye Temperatures of the Constituents of a Composite Superconductor. Physica B: Condensed Matter, 628, Article ID: 413559. https://doi.org/10.1016/j.physb.2021.413559
[16]	Zhao, G. (2002) Unconventional Phonon-Mediated Superconductivity in MgB2. New Journal of Physics, 4, 3. https://doi.org/10.1088/1367-2630/4/1/303
[17]	Leggett, A.J. (2018) PHYS598/2 Lecture 2: Non-Cuprate Exotics I: BKBO, MgB2, Alkali Full-Erides (courses.physics.illinois.edu).
[18]	Ramshaw, B.J., Sebastian, S.E., McDonald, R.D., Day, J., Tan, B.S., Zhu, Z., et al. (2015) Quasiparticle Mass Enhancement Approaching Optimal Doping in a High-tc Superconductor. Science, 348, 317-320. https://doi.org/10.1126/science.aaa4990
[19]	Di Gioacchino, D., et al. (1997) Frequency Dependence of Susceptibility Higher Harmonics in YBCO Sample by Numerical Solutions of the Non-Linear Magnetic Flux Diffusion Equation. Physica C: Superconductivity and Its Applications, 282-287, 2157-2158. https://doi.org/10.1016/S0921-4534(97)01203-3
[20]	Alexandrov, A.S. (2001) Nonadiabatic Polaronic Superconductivity in MgB2 and Cuprates. Physica C: Superconductivity, 363, 231-236. https://doi.org/10.1016/s0921-4534(01)01095-4
[21]	Dogan, M. and Cohen, M.L. (2021) Anomalous Behavior in High-Pressure Carbonaceous Sulfur Hydride. Physica C: Superconductivity and its Applications, 583, Article ID: 1353851. https://doi.org/10.1016/j.physc.2021.1353851
[22]	Talantsev, E.F. (2022) Universal Fermi Velocity in Highly Compressed Hydride Superconductors. Matter and Radiation at Extremes, 7, Article ID: 058403. https://doi.org/10.1063/5.0091446
[23]	Minkov, V.S., Ksenofontov, V., Bud’ko, S.L., Talantsev, E.F. and Eremets, M.I. (2023) Magnetic Flux Trapping in Hydrogen-Rich High-Temperature Superconductors. Nature Physics, 19, 1293-1300. https://doi.org/10.1038/s41567-023-02089-1
[24]	Malik, G.P. and Varma, V.S. (2023) Compressed H3S: Fits to the Empirical Hc2(t) Data and a Discussion of the Meissner Effect. World Journal of Condensed Matter Physics, 13, 111-127. https://doi.org/10.4236/wjcmp.2023.134008
[25]	Malik, G.P. (2024) An Explanation of the Temperature-Dependent Upper Critical Field Data of H3S on the Basis of the Thermodynamics of a Superconductor in a Magnetic Field. World Journal of Condensed Matter Physics, 14, 45-50. https://doi.org/10.4236/wjcmp.2024.143005