An Inquiry into Two Intriguing Values of the Critical Current Density of Bi-2212

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1. Introduction

Among the family of Bi-based cuprates, Bi_{2}Sr_{2}CaCu_{2}O_{8} (Bi-2212) and Bi_{2}Sr_{2}Ca_{2}Cu_{3}O_{10} (Bi-2223) belong to the class of superconductors (SCs) that has the highest values of the critical temperature (*T _{c}*), critical magnetic field (

Employing the formalism of the Bethe-Salpeter equation (BSE), we deal in this paper with two empirical values of *j _{c}* (

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

A feature that we did not pay heed to while dealing with *j _{c}*(

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.

2. Theoretical Framework

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):

$\int \text{d}{p}_{x}\text{d}{p}_{y}}=2\pi eH{\displaystyle \underset{n}{\sum}\text{\hspace{0.05em}}},\text{}\frac{{p}_{x}^{2}}{2{m}^{*}}+\frac{{p}_{y}^{2}}{2{m}^{*}}=\left(n+1/2\right)\hslash \Omega \left(H,\eta \right)$

${\Omega}_{0}=1.7588\times {10}^{7}\text{\hspace{0.05em}}\text{rad}\cdot {\text{sec}}^{-1}\cdot {\text{G}}^{-1}\text{}\left(\text{for a particle of mass}={m}_{e}\right)$ (1)

$\Omega \left(H,\eta \right)={\Omega}_{0}H/\eta \text{}\left(\text{for a particle of mass}={m}^{*}=\eta {m}_{e}\right)$

where *m ^{*}* is the effective mass of a charge carrier and

$\left(2/3\right)\left(\mu -k\theta \right)\le \left({p}_{x}^{2}+{p}_{y}^{2}\right)/2{m}^{*}=\left(n+1/2\right)\Omega \left({H}_{c},\eta \right)\le \left(2/3\right)\left(\mu +k\theta \right)$

$\left(1/3\right)\left(\mu -k\theta \right)\le {p}_{z}^{2}/2{m}_{1}^{*}\le \left(1/3\right)\left(\mu +k\theta \right)$.

We note that in lieu of the second of the above relations, we had *incorrectly* employed (
$-k\theta /3\le {p}_{z}^{2}/2{m}^{*}-\mu \le k\theta /3$ ) 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
$\left\{eV,\hslash =1,c=1\right\}$.

2.1. The Pairing Equation Incorporating *μ*, *T* and *H* When *P* = 0 and Hence *j _{c}* = 0

Denoting the values of *j _{c}*, the magnetic interaction parameter

$\begin{array}{l}1={\lambda}_{m0}{\displaystyle {\int}_{{L}_{1}}^{{L}_{{}_{2}}}\frac{\text{d}\xi}{\sqrt{1+\xi /{\mu}_{0}}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\times {\displaystyle \underset{n=0}{\overset{{n}_{1}\left(\theta ,\rho ,{H}_{c0},{\eta}_{0}\right)}{\sum}}\frac{\mathrm{tanh}\left[\left(\Omega \left({H}_{c0},{\eta}_{0}\right)/2k{T}_{c0}\right)\left\{\xi /\Omega \left({H}_{c0},{\eta}_{0}\right)+n+1/2\right\}\right]}{\Omega \left({H}_{c0},{\mu}_{0}\right)\left[\xi /\Omega \left({H}_{c0},{\eta}_{0}\right)+n+1/2\right]}},\end{array}$ (2)

where

${\lambda}_{m0}=\frac{e{H}_{c0}V}{16{\pi}^{2}}\sqrt{\frac{2{\eta}_{0}{m}_{e}}{\rho k\theta}}$

${L}_{1}=-\left(2\rho +1\right)k\theta /3,\text{}{L}_{2}=\left(-2\rho +1\right)k\theta /3$

${n}_{1}\left(\theta ,\rho ,{H}_{c0},{\eta}_{0}\right)=floor\left[\frac{2\left(\rho +1\right)k\theta}{3\Omega \left({H}_{c0},{\eta}_{0}\right)}-1/2\right]$

${\mu}_{0}=\rho k\theta $

and, as shown in Paper I, *λ _{m}*

Remarks:

1) Equation (2) above is identical with Equation (5) of Paper I* except* that, because of the error in the application of LEE, the limits of the integral in the latter equation were from—*L*_{1}(*θ*) to *L*_{1}(*θ*) and the upper limit of the sum was *n _{m}*(

${L}_{1}\left(\theta \right)=k\theta /3,{n}_{m}\left(\theta ,{H}_{c}\right)=floor\left\{\frac{2k\theta}{3\hslash {\Omega}_{1}\left({H}_{c}\right)}-\frac{1}{2}\right\}.$

2) A more compact form of (2) given in Paper II that we employ here is

$\begin{array}{l}1=2{\lambda}_{m0}{\displaystyle {\int}_{{L}_{3}\left(\rho \right)}^{{L}_{4}\left(\rho \right)}\text{d}z}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\times {\displaystyle \underset{n=0}{\overset{{n}_{1}\left(\theta ,\rho ,{H}_{c1},{\eta}_{1}\right)}{\sum}}\frac{\mathrm{tanh}\left[\left(\rho \theta /2{T}_{c0}\right)\left\{{z}^{2}-1+\left(n+1/2\right)\Omega \left({H}_{c1},{\eta}_{0}\right)/\rho k\theta \right\}\right]}{{z}^{2}-1+\left(n+1/2\right)\Omega \left({H}_{c0},{\eta}_{0}\right)/\rho k\theta}}\end{array}$ (3)

where

${L}_{3}\left(\rho \right)=\sqrt{\left(\rho -1\right)/3\rho},\text{}{L}_{4}\left(\rho \right)=\sqrt{\left(\rho +1\right)/3\rho}\text{.}$

2.2. The Pairing Equation Incorporating *μ*, *T* and *H* When *P* ≠ 0 and Hence *j _{c}* ≠ 0

For this case, in lieu of the (erroneous) Equation (17) in Paper I, we have the following compact equation where* μ _{i}* has been employed interchangeably with

$1={\lambda}_{mi}{\displaystyle \underset{{z}_{Li}}{\overset{{z}_{Ui}}{\int}}\text{d}z}{\displaystyle \underset{n=0}{\overset{{n}_{L}}{\sum}}\frac{\mathrm{tanh}\left[{A}_{i}\right]+\mathrm{tanh}\left[{B}_{i}\right]}{{z}^{2}-1+\left(n+1/2\right)\hslash \Omega \left({H}_{ci},{\eta}_{i}\right)/{\mu}_{i}}},\text{}\left(i=1\text{or}2\right)$ (4)

where

${\lambda}_{mi}=\frac{{H}_{ci}}{{H}_{c0}}\sqrt{\frac{{\eta}_{i}{\mu}_{0}}{{\eta}_{0}{\mu}_{i}}}{\lambda}_{m0}$ (5)

${z}_{Li}\text{}=\sqrt{1+\frac{-1-2{q}_{i}\rho +1/{y}_{i}}{3{q}_{i}\rho}},\text{}{z}_{Ui}=\sqrt{1+\frac{1-2{q}_{i}\rho -1/{y}_{i}}{3{q}_{i}\rho}}$

${n}_{Li}=floor\left[\frac{2k\theta}{3\hslash \Omega \left({H}_{ci},{\eta}_{i}\right)}\left(1+{q}_{i}\rho -1/{y}_{i}\right)-1/2\right]$ (6)

$\begin{array}{l}\mathrm{tanh}\left[{A}_{i}\right]=\mathrm{tanh}\left[\left({q}_{i}\rho \theta /2{T}_{ci}\right)\left\{{z}^{2}-1+\left(n+1/2\right)\Omega \left({H}_{ci},{\eta}_{i}\right)/{q}_{i}\rho k\theta +1/{q}_{i}\rho {y}_{i}\right\}\right]\\ \mathrm{tanh}\left[{B}_{i}\right]=\mathrm{tanh}\left[\left({q}_{i}\rho \theta /2{T}_{ci}\right)\left\{{z}^{2}-1+\left(n+1/2\right)\Omega \left({H}_{ci},{\eta}_{i}\right)/{q}_{i}\rho k\theta -1/{q}_{i}\rho {y}_{i}\right\}\right]\end{array}$

${y}_{i}=\frac{k\theta}{\left|{P}_{i}\right|}\sqrt{\frac{6{m}_{i}^{*}}{{\mu}_{i}}}\text{}\left(\left|{P}_{i}\right|\text{and}{m}_{1}^{*}\text{in units of electron-Volt}\right)$ (7)

and the subscript *i* = 1 denotes that the symbol corresponds to *j _{c}*

It is easily seen that when *T _{ci}* =

Equation (4) does not shed light on the relative contributions of the *e-e* and the* h-h* pairs to it, which is an additional feature that we are now interested in. This is a situation that is easily remedied by revisiting the derivation of (4), the starting point of which was Equation (9) in Paper II in which the limits of the integral (say *F*) were from

$\left(\mu -k\theta +\alpha \right)/3\text{to}\left(\mu +k\theta -\alpha \right)/3\text{}\left(\alpha =\left|P\right|\sqrt{\mu /6{m}^{*}{c}^{2}}\right)\text{.}$

If we split these limits into two parts as from
$\left(\mu -k\theta +\alpha \right)/3$ to
$\mu /3$ and from
$\mu /3$ to
$\left(\mu +k\theta -\alpha \right)/3$, change the variable of integration from *p _{z}* to
$\xi ={p}_{z}^{2}/2m-\mu /3$ and follow it up just as we did in Paper II, then

$1={F}_{1i}\left({\lambda}_{m0},\theta ,\rho ,{\mu}_{i},{T}_{c1},{H}_{ci},{y}_{i}\right)+{F}_{2i}\left({\lambda}_{m0},\theta ,\rho ,{\mu}_{i},{T}_{c1},{H}_{ci},{y}_{i}\right),$ (8)

where

$\begin{array}{l}{F}_{1i}\left({\lambda}_{m0},\theta ,\rho ,{\mu}_{i},{T}_{c1},{H}_{ci},{y}_{i}\right)\\ ={\lambda}_{mi}{\displaystyle \underset{{L}_{i}\left(\rho ,{\mu}_{i},{y}_{i}\right)}{\overset{1/\sqrt{3}}{\int}}\text{d}z}{\displaystyle \underset{n=0}{\overset{{n}_{L}}{\sum}}\frac{\mathrm{tanh}\left[{A}_{i}\right]+\mathrm{tanh}\left[{B}_{i}\right]}{{z}^{2}-1+\left(n+1/2\right)\hslash \Omega \left({H}_{ci},{\eta}_{i}\right)/{q}_{i}\rho k\theta}}\end{array}$ (9)

$\begin{array}{l}{F}_{2i}\left({\lambda}_{m0},\theta ,\rho ,{\mu}_{i},{T}_{c1},{H}_{ci},{y}_{i}\right)\\ ={\lambda}_{mi}{\displaystyle \underset{1/\sqrt{3}}{\overset{{U}_{i}\left(\rho ,{\mu}_{i},y\right)}{\int}}\text{d}z}{\displaystyle \underset{n=0}{\overset{{n}_{L}}{\sum}}\frac{\mathrm{tanh}\left[{A}_{i}\right]+\mathrm{tanh}\left[{B}_{i}\right]}{{z}^{2}-1+\left(n+1/2\right)\hslash \Omega \left({H}_{ci},{\eta}_{i}\right)/{q}_{i}\rho k\theta}},\end{array}$ (10)

and

${L}_{i}\left(\rho ,{\mu}_{i},{y}_{i}\right)=\mathrm{Re}\left[\sqrt{\frac{1}{3}-\frac{1-1/{y}_{i}}{3{\mu}_{i}/k\theta}}\right],\text{}{U}_{i}\left(\rho ,{\mu}_{i},{y}_{i}\right)=\left[\sqrt{\frac{1}{3}+\frac{1-1/{y}_{i}}{3{\mu}_{i}/k\theta}}\right].$

When (8) is solved in conjunction with the number equation given below, (9) and (10), respectively, give the desired relative contributions of the *h-h *and the *e-e* pairs to *j _{c}*

2.3. The *T*- and *H*-Incorporated Number Equation

In lieu of the naïve equation employed in Paper I

${n}_{s}\left({E}_{F}\right)=\frac{1}{3{\pi}^{2}}{\left[\frac{2{m}^{*}{E}_{F}}{{\left(\hslash c\right)}^{2}}\right]}^{3/2}\text{}\left({m}^{*}\text{and}{E}_{F}\text{in units of electron-Volt}\right)\text{,}$

the number equation employed in Paper II is

$\begin{array}{l}{n}_{s}{}_{i}\left(\theta ,{\mu}_{i},{T}_{ci},{H}_{ci},{\eta}_{i}\right)=2{P}_{F}\left({H}_{c}{}_{i}\right){\displaystyle {\int}_{0}^{{x}_{U}}\text{d}x}\\ \times {\displaystyle \underset{n=0}{\overset{{n}_{m}\left(\theta ,{\mu}_{i},{H}_{ci},{\eta}_{i}\right)}{\sum}}\left[1-\mathrm{tanh}\left\{\frac{\hslash \Omega \left({H}_{ci},{\eta}_{i}\right)}{2k{T}_{ci}}\left[n+1/2+{x}^{2}-\frac{{\mu}_{i}}{\hslash \Omega \left({H}_{ci},{\eta}_{i}\right)}\right]\right\}\right]}\text{\hspace{0.05em}}\text{\hspace{0.05em}},\end{array}$ (11)

where

${P}_{F}\left({H}_{c}{}_{i}\right)=\frac{{b}_{1}{b}_{2}{H}_{c}{}_{i}}{4\sqrt{2}{\pi}^{2}}\sqrt{\frac{{m}_{e}{c}^{2}{\Omega}_{0}{H}_{ci}}{{a}_{3}}}\frac{1}{{\left(\hslash c\right)}^{3}},\text{}{x}_{U}=\sqrt{\frac{{\mu}_{i}+k\theta}{3\hslash \Omega \left({H}_{c}{}_{i},{\eta}_{i}\right)}}$

$\begin{array}{l}{b}_{1}={\left(137.0359895\right)}^{-1/2},\text{}{b}_{2}=6.9250774\times {10}^{-2},\text{}{a}_{3}=1.51926689\times {10}^{15}\\ {n}_{m}\left(\theta ,{\mu}_{i},{H}_{ci},\eta \right)=floor\left[\frac{2\left(k\theta +{\mu}_{i}\right)}{3\hslash \Omega \left({H}_{c}{}_{i},{\eta}_{i}\right)}-1/2\right].\end{array}$

Equation (11) was derived in [5].

From the definitions of *y _{i} *given below (4) and
$\left|{P}_{ci}\right|=2{m}_{i}^{*}\left|{v}_{c}{}_{i}\right|$, it follows that

$\left|{v}_{ci}\left(\theta ,\rho ,{q}_{i},{\eta}_{i},{y}_{i}\right)\right|=\frac{c}{2{y}_{i}}\sqrt{\frac{6k\theta}{{q}_{i}\rho {\eta}_{i}\left({m}_{e}{c}^{2}\right)}}\text{,}$ (12)

whence substituting (11) and (12) into ${j}_{ci}\left({T}_{ci},{H}_{ci}\right)=e{n}_{si}\left({T}_{ci},{H}_{ci}\right){v}_{ci}\left({T}_{ci},{H}_{ci}\right)$, we obtain

$1-\frac{{j}_{ci}\left({T}_{ci},{H}_{ci}\right)}{e{n}_{si}\left(\theta ,{q}_{i},{T}_{ci},{H}_{ci},{\eta}_{i}\right)\left|{v}_{ci}\left(\theta ,\rho ,{q}_{i},{\eta}_{i},{y}_{i}\right)\right|}=0.$ (13)

3. Dealing with the “Intriguing” Values of* j _{c}* of Bi-2212

To deal with the “intriguing” values of *j _{c}* of Bi-2212, we also need empirical data corresponding to the

$\theta \left(\text{Caions}\right)=\text{237K}$

${T}_{c0}=65\text{K},\text{}{H}_{c0}=\text{36}\times {\text{10}}^{4}\text{G},\text{}{j}_{c0}=0$ (14)

${T}_{c1}=4.2\text{K},\text{}{H}_{c1}=12\times {10}^{4}\text{G},\text{}{j}_{c1}=2.4\times 1{0}^{5}\text{A}/{\text{cm}}^{2}$ (15)

${T}_{c2}={T}_{c1},\text{}{H}_{c2}={H}_{c1},\text{}{j}_{c2}=1.0\times 1{0}^{6}\text{A}/{\text{cm}}^{2}.$ (16)

Taking stock of the unknown parameters in our problem, we find that we have five of them viz.,* ρ* (or *μ*_{0}),* η*_{0}, *q _{i}* (or

3.1. A Template for the Values of Various Parameters Associated with* j _{c}*

Guided by the current wisdom about the values of some select parameters in the superconductivity of high-*T _{c}* SCs, such as:

1) they are characterized by much lower values of the Fermi energy or the chemical potential than the elemental SCs for which it is of the order of 1 - 10 eV;

2) the effective mass of electrons in one of them, viz., MgB_{2}, has been estimated to be 0.44 - 0.68 [6];

3) the number density of charge-carriers (*n _{s}*) in these SCs is several orders of magnitude smaller than for the elemental SCs for which it is of the order of 1.0 × 10

4) the charge-carriers in some of them are *e-e* pairs, while for some others they are *h-h* pairs, we adopt *approximately* the following values as a template in this study.

a) *μ*_{0}, *μ*_{1}, *μ*_{2} < 200 meV

b) 0.4 < *η*_{0}, *η*_{1}, *η*_{2} < 1.4

c) *n _{s}*

d) *λ _{m}*

e) *F*_{1} > 50% (*F*_{2} > 50%) for* j _{ci}* to be regarded as predominantly due to

3.2. Procedure and Results

Our procedure comprises the following steps:

1) Solve the core equation (2) for *λ _{m}*

2) The above step implies that one has on command an innumerable number of the triplets of {*μ*_{0}, *η*_{0}, *λ _{m}*

3) As an illustration of the above remark, we give below—both for* j _{c}*

a) *S*_{1} (*μ*_{0}, *η*_{0}, *λ _{m}*

b) *S*_{2} (*μ*_{0}, *η*_{0}, *λ _{m}*

c) *S*_{3} (*μ*_{0}, *η*_{0}, *λ _{m}*

4) Given in Table 1 are the detailed results corresponding to {*j _{c}*

5) Although we have obtained results corresponding to {*j _{c}*

Table 1. Treating *μ*_{1} as the independent variable, values of *η*_{1} and *y*_{1} obtained by solving (8) and (13) with the inputs of S_{1} (*μ*_{0}, *η*_{0}, *λ _{m}*

Table 2. Following from the sets {*j _{c}*

4. Discussion

1) A feature of the study reported here that is worth drawing attention to is that, unlike the *T _{c}*s 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}\gg k\theta $. 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*
$\gg k\theta $, 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 _{c}*

i) *μ*_{1} < 16.0, 11.0, 19.1 (meV)

ii) *n _{s}*

4) If it is empirically found that the charge-carriers corresponding to* j _{c}*

i) *μ*_{1} (meV): (6.0 - 16.0), (7.0 - 11.0), (6.0 - 19.1)

ii) *n _{s}* (×10

5) If the charge carriers are predominantly *h-h* pairs, then the widths of the values *μ*_{2} and* n _{s}*

i) *μ*_{2} (meV): (7.80 - 27.0), (8.10 - 38.0), (7.40 - 24.5)

ii) *n _{s}*

5) Corresponding to the values of *μ*_{1} and *μ*_{2} in 3) and 4) above, the widths of the values of *η*_{1} and *η*_{2}, 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 *μ*_{1} and *μ*_{2} have values that are significantly lower than the values for the elemental SCs, *μ*_{2} > *μ*_{1}.

i) Unsurprisingly,* n _{s}*

ii) While *η*_{1} decreases as* μ*_{1} is increased, as does *η*_{2} when *μ*_{2} is increased, quite generally, *η*_{2} > *η*_{1}, implying that* j _{c}*

5. Conclusions

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

Since, as was noted above, there is an innumerable number of the triplets of (*μ*_{0}, *η*_{0}, *λ _{m}*

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 _{c}*

Parameters characterizing *j _{c}*

${\mu}_{1}=12.3\text{\hspace{0.17em}}\text{meV},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\eta}_{1}=0.584,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{s1}=1.01\times {10}^{18}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{cm}}^{-3}.$

Parameters characterizing *j _{c}*

${\mu}_{2}=22.7\text{\hspace{0.17em}}\text{meV},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\eta}_{2}=0.944,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{s2}=3.55\times {10}^{18}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{cm}}^{-3}.$

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.

References

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