Among the family of Bi-based cuprates, Bi₂Sr₂CaCu₂O₈ (Bi-2212) and Bi₂Sr₂Ca₂Cu₃O₁₀ (Bi-2223) belong to the class of superconductors (SCs) that has the highest values of the critical temperature (T_c), critical magnetic field (H_c) and critical current density (j_c). Although Bi-2223 is characterized by T_c ≈ 110 K and Bi-2212 by T_c ≈ 95 K, the former—unlike the latter—has not only been found very hard to fabricate, but also almost impossible to obtain in a predominantly single phase. For these reasons, Bi-2212 is an SC that is among those that are most suitable for industrial applications and hence—given that as yet there is no consensus about the detailed mechanism that leads to high-T_cs—it warrants that its superconducting features be probed from as many different approaches as possible.

Employing the formalism of the Bethe-Salpeter equation (BSE), we deal in this paper with two empirical values of j_c (T, H), i.e., 2.4 × 10⁵ and 1.0 × 10⁶ A/cm² which are intriguing because they have been reported for the same values of T and H, viz T = 4.2 K and H = 12 × 10⁴ G [1]. This is a problem that we had earlier addressed in [2] —Paper I hereafter, based on the novel premise that the chemical potential (μ) of an SC subsumes most of the features to which its widely varying values of j_c(T, H) are conventionally attributed—features such as its geometry (wire, tape, thin film, etc.) and dimensions, the type of dopants it contains, and the manner of its preparation which causes different samples of it to be characterized by different granular structures and grain boundaries, alignment of the grains and so on, see e.g., [3], most of which are not quantifiable. Another remarkable feature of the approach followed in Paper I is that, unlike almost all conventional approaches, it employs directly the basic equation that defines j_c(T, H) as the product of the electronic charge e, the number of superconducting charge-carriers n_s(T, H) and their critical velocity v_c(T, H).

The framework of Paper I comprised three core equations: (i) a μ-, T- and H-dependent pairing equation corresponding to P = 0 and hence j_c = 0, where P denotes the 3-momentum of Cooper pairs in the lab frame, (ii) a μ-, T- and H-dependent pairing equation corresponding to P ≠ 0 and hence j_c ≠ 0 and (iii) an equation relating Fermi energy E_F with the number density of the charge-carriers n_s. It was pointed out in [4] —Paper II hereafter, which was devoted to a study of the variation of j_c(T, H) with T (for a fixed value of H) and H (for a fixed value of T) that the framework of Paper I was deficient in that 1) the limits of both the P = 0 and the P ≠ 0 equations were incorrect for a reason that will be given below, 2) the assumption that P = 0 and P ≠ 0 equations are characterized by the same interaction parameter is unjustifiable and hence should be done away with and 3) it employed an equation for n_s which, strictly speaking, is valid only when both T = 0 and H = 0, in lieu of which a more accurate μ-, T- and H-dependent number equation was employed in Paper II.

A feature that we did not pay heed to while dealing with j_c(T, H) values of both the elemental and composite SCs in several papers (Refs. 6 - 11 in Paper II) is the relative contributions of the electron-electron (e-e) and the hole-hole (h-h) pairs to j_c. We deal here with the problem of the “intriguing” values of j_c(T, H) of Bi-2212 in the enlarged framework of Paper II which now includes this feature.

The paper is organized as follows. Given in the next section is the theoretical framework employed in this study. In essence, it is a slightly enhanced framework of Paper II in that it now distinguishes between the two types of pairs as noted above. Dealt with in detail in Section 3 is the problem of the “intriguing” values of j_c. Sections 4 and 5, respectively, are devoted to a Discussion of our findings and the Conclusions following from them.

Both in Paper I and Paper II, the μ- and T-incorporated BSE was further generalized to include H applied in the z-direction via the following substitutions which constitute the Landau quantization prescription (LQP):

∫ d p x d p y = 2 π e H ∑ n   , p x 2 2 m * + p y 2 2 m * = ( n + 1 / 2 ) ℏ Ω ( H , η )

Ω 0 = 1.7588 × 10 7   rad ⋅ sec − 1 ⋅ G − 1 ( for a particle of mass = m e ) (1)

Ω ( H , η ) = Ω 0 H / η ( for a particle of mass = m * = η m e )

where m^* is the effective mass of a charge carrier and m_e the free electron mass. The upper limits of the integral and sum in (1) are usually infinity, which cannot be so in our problem because the energy of an electron/hole is constrained to lie between μ − k θ and μ + k θ , where k is the Boltzmann constant and θ the Debye temperature of the ions. Hence, its energy was divided between the (p_x + p_y) and p_z degrees of freedom by appealing to the law of equipartition of energy (LEE) as

( 2 / 3 ) ( μ − k θ ) ≤ ( p x 2 + p y 2 ) / 2 m * = ( n + 1 / 2 ) Ω ( H c , η ) ≤ ( 2 / 3 ) ( μ + k θ )

( 1 / 3 ) ( μ − k θ ) ≤ p z 2 / 2 m 1 * ≤ ( 1 / 3 ) ( μ + k θ ) .

We note that in lieu of the second of the above relations, we had incorrectly employed ( − k θ / 3 ≤ p z 2 / 2 m * − μ ≤ k θ / 3 ) in Paper I.

Recapitulated below from Paper II are the equations which comprise our framework in this study. For the convenience of the reader, they have been written in terms of the units commonly employed in the BCS theory, e.g., Gauss for H, although they were derived by employing the natural system of units { e V , ℏ = 1 , c = 1 } .

To deal with the “intriguing” values of j_c of Bi-2212, we also need empirical data corresponding to the j_c = 0 situation in order to fix the interaction parameters when j_c ≠ 0. Assuming as in Paper II that the pairing mechanism is the 1-phonon exchange mechanism due to Ca ions, given below are the empirical data [1] that we have employed for both these cases in our study.

θ ( Caions ) = 237K

T c 0 = 65 K , H c 0 = 36 × 10 4 G , j c 0 = 0 (14)

T c 1 = 4.2 K , H c 1 = 12 × 10 4 G , j c 1 = 2.4 × 1 0 5 A / cm 2 (15)

T c 2 = T c 1 , H c 2 = H c 1 , j c 2 = 1.0 × 1 0 6 A / cm 2 . (16)

Taking stock of the unknown parameters in our problem, we find that we have five of them viz., ρ (or μ₀), η₀, q_i (or μ_i), η_i and y_i (because fixing ρ and η₀ fixes λ_m0, whence λ_mi too is fixed), whereas the number of equations at our disposal is three viz. (2), (8) and (13). It follows that we need to assume the values of two of these parameters in order to calculate the rest of them. Based on diverse criteria, we give below a template for the values of various parameters and demand that both types of parameters conform to it.

1) A feature of the study reported here that is worth drawing attention to is that, unlike the T_cs and the gaps of Bi-2212 which require 2- and 3-phonon exchange mechanisms [7], we needed here only the one-phonon exchange mechanism. This of course is due to the fact that putting an SC in a magnetic field considerably weakens the interaction parameter.

2) Although the role of the e-e and the h-h pairs in the original BCS theory has been a matter of debate in some quarters, it has been shown [8] that both types of pairs make equal contributions to the pairing amplitude when T = 0, H = 0 and E F ≫ k θ . The fact that we identified the pairs below the chemical potential surface as h-h pairs and those above it as e-e pairs is because, as has also been noted by Ziman [9], “the excitations of the superconducting state are peculiar quasi-particles which change from being ‘electrons’ to being ‘holes’ as they pass through the Fermi level”. Since we are now dealing with the situation where T ≠ 0, H ≠ 0 via LQP (which brings the effective mass of the charge-carriers into play) and values of μ that are not ≫ k θ , unsurprisingly, we have found that the two types of carriers, in general, do not make equal contributions. What seems remarkable is that we can quantify the contributions of such pairs in the mean-field approximation, i.e., without taking recourse to a Hamiltonian that has explicit terms corresponding to the e-e and the h-h pairs—which is an approach that been advocated by Llano and collaborators in several papers as, e.g., [10].

3) In dealing with j_c1 subject to the constraints noted in Section 3.1, we have found that the values of μ₁ and n_s1—which may be empirically verifiable—corresponding to S₁, S₂ and S₃, respectively, are:

i) μ₁ < 16.0, 11.0, 19.1 (meV)

ii) n_s1 ≤ 11.6, 6.65, 15.4 (×10¹⁷ cm⁻³)

4) If it is empirically found that the charge-carriers corresponding to j_c1 are predominantly h-h pairs, then the widths of the values of μ₁ and n_s1, corresponding to S₁, S₂ and S₃, respectively, are further narrowed down to

i) μ₁ (meV): (6.0 - 16.0), (7.0 - 11.0), (6.0 - 19.1)

ii) n_s (×10¹⁷ cm⁻³): (6.76 - 11.6), (5.63 - 6.65), (7.05 - 15.4)

5) If the charge carriers are predominantly h-h pairs, then the widths of the values μ₂ and n_s2 similar to those in 3) and corresponding to {j_c2, S₁}, {j_c2, S₂} and {j_c2, S₃), respectively, are:

i) μ₂ (meV): (7.80 - 27.0), (8.10 - 38.0), (7.40 - 24.5)

ii) n_s2 (×10¹⁸ cm⁻³): (2.35 - 4.94), (2.10 - 5.11), (2.40 - 4.95).

5) Corresponding to the values of μ₁ and μ₂ in 3) and 4) above, the widths of the values of η₁ and η₂, respectively, are: (0.7434 - 0.4050), (1.4015 - 0.4030), (0.7654 - 0.4146) and (1.3105 - 0.6553), (1.1337 - 0.4888), (1.4000 - 0.7178).

6) While both μ₁ and μ₂ have values that are significantly lower than the values for the elemental SCs, μ₂ > μ₁.

i) Unsurprisingly, n_s2 > n_s1 and (mostly) v_c2 < v_c1.

ii) While η₁ decreases as μ₁ is increased, as does η₂ when μ₂ is increased, quite generally, η₂ > η₁, implying that j_c2 is due to heavier charge carriers, in the context of which it seems relevant to point out that one is dealing here with a non-equilibrium state when the system is subject to an external stimulus.

It was noted in the Introduction above that there is as yet no consensus about the cause of the occurrence of high-T_c superconductivity although, as expounded in, e.g. [11], a wide variety of mechanisms or models—such as the Nambu-Eliashberg-McMillan extension of the BCS theory, non-phononic mechanisms invoking magnons (the Hubbard model, the t-J model), the RVB model, electronic models invoking excitons, and so on—has been proposed to account for it. Given such a situation, it is warranted that due heed be paid to the insights that all these approaches provide. Obtained via the BSE-based approach, we deem it plausible that the results reported in this paper such as the values of the chemical potential and the effective masses of the charge-carriers corresponding to the j_c0, j_c1 and j_c2 may aid in reaching the goal of the final theory of high-T_c superconductivity.

Since, as was noted above, there is an innumerable number of the triplets of (μ₀, η₀, λ_m0) which can be employed for determining the values of various parameters associated with j_ci, it is remarkable that both the values of μ_i and n_si corresponding to j_ci (i = 1 or 2) fall in nearly the same ballpark for each of the three triplets S₁, S₂ and S₃ that we have employed. This implies that the values of various parameters corresponding to j_ci—not restricted to μ_i and n_si—depend only mildly on the triplets S₁ - S₃ employed to obtain them and that by appealing to just a few of them one is led to better than an order-of-magnitude estimate of these parameters.

If the width of each parameter in Table 2 is replaced by its mid-point, which is not unreasonable, then the values of some of the parameters that distinguish j_c1 and j_c2 when the charge-carriers are predominantly h-h pairs are as follows:

Parameters characterizing j_c1:

μ 1 = 12.3   meV ,     η 1 = 0.584 ,     n s 1 = 1.01 × 10 18     cm − 3 .

Parameters characterizing j_c2:

μ 2 = 22.7   meV ,     η 2 = 0.944 ,     n s 2 = 3.55 × 10 18     cm − 3 .

It is gratifying to note that our suggestion that regardless of the physical attributes of an SC, its Fermi energy (i.e., μ at T = 0) plays an important role in determining its properties is being taken note of, as evidenced by [12].

We conclude by drawing attention to [13] for an exposition of the BSE-based approach to superconductivity.

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

Malik, G.P. and Varma, V.S. (2021) An Inquiry into Two Intriguing Values of the Critical Current Density of Bi-2212. World Journal of Condensed Matter Physics, 11, 53-64. https://doi.org/10.4236/wjcmp.2021.113004