**1. Introduction**

### **1.1 A historical note**

The phenomenon of superconductivity was first observed by Kamerlingh Onnes [1] in 1911 while studying the electrical resistivity of metallic mercury as a function of temperature when cooled to liquid helium temperatures. He noticed that at the critical temperature Tc = 4.2 K (269°C), the resistance vanished abruptly. This was an exciting discovery because it suggested that such materials could have immense practical applications. Since cooling a material to very low temperatures is a tedious and expensive process, Onne's discovery triggered a

search in scientific laboratories across the world for other superconducting materials having Tcs higher than that of Hg, the ultimate aim being to find a material which is superconducting at room temperature. While this search led to the discovery of many elements and alloys that are superconductors (SCs), till as late as in the 1980s, the highest Tc of any known SC was about 23 K (for Nb3Sn). A radical change occurred in this situation in 1986 when Bednorz and Müller announced the discovery of an SC (Ba-La-Cu-O system) with *Tc* ≈38 K*,*for which they were awarded the Nobel Prize in 1987. An avalanche of activity followed soon thereafter leading to the discovery of many "high-temperature SCs" such as MgB2 ð Þ *Tc* ≃40 K , Fe-based SCs highest ð Þ *Tc* ≃ 55 K *,* and "cuprates," examples of which are Bi-, Tl-, and Hg-based SCs that contain one or more units of CuO2 as a constituent, the Tcs of these SCs at ambient pressure being about 95, 110, and 138 K, respectively, and for the Hg-based SCs, going up to 164 K at high pressure. The confirmed current record for the highest Tc (203 K; at a pressure of 150 GPa) is held by H3S which was discovered in 2015. It would thus appear that realization of the dream of room temperature SCs is near at hand.

Matsubara [4] in 1955. Not so well-known is the third development that took place in 1954 due to Okubo [5], viz., the introduction of the concept of a superpropagator as the propagator for the non-polynomial field exp (gϕ) where g is a coupling

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features…*

BSE is a four-dimensional equation which was invented for the description of bound states of two relativistic particles, hordes of which were being discovered in the late 1940s and early 1950s in cosmic rays and high-energy accelerators around the world. A general perception about the BSE is that it is irrelevant for supercon-

FTFT is obtained via an adaption of quantum field theory where time is replaced by temperature as the variable in terms of which evolution of a system is studied. Because temperature is a statistical concept, this is a huge step which converts an equation setup for the bound states of two particles interacting in vacuum into an equation valid in a many-body system. A shortcut known as the Matsubara prescription or recipe for introducing temperature into a BSE is to replace in it the

*q*<sup>4</sup> ¼ ð Þ 2*n* þ 1 *π=*ð Þ �*iβ* for fermions

¼ 2n*π=*ð Þ �*iβ* for bosons

Okubo's work remained an obscure academic exercise till Salam and collaborators found that non-polynomial theories have inbuilt damping effects. It was therefore hoped that they might provide the means of renormalizing, e.g., the weak interactions. These theories were not pursued any further after the gauge theory of interactions became widely accepted as the correct theory. However, it is relevant in the context of superconductivity to note that the expression for the Feynman superpropagator corresponding to the non-polynomial interaction 1*=*ð Þ 1 þ *gϕ* is [7]:

∞ð

*dte*�*<sup>t</sup>*

*k*2

ð Þ 2 � *t*

*<sup>g</sup>*<sup>2</sup> <sup>þ</sup> *<sup>t</sup> :* (2)

*<sup>k</sup>*<sup>2</sup> � *<sup>i</sup><sup>ε</sup>* (3)

0

The comparison of the above with the expression for one-particle Feynman

*<sup>F</sup>* ð Þ¼ *<sup>k</sup>* <sup>1</sup>

reveals that a superpropagator represents a weighted superposition of multiple quanta. The significance of this remark emerges if we recall how the electron-latticeelectron interaction in an elemental SC causes formation of CPs via 1PEM. Appealing to the same picture for a composite SC, it follows that pairing can now also be caused via more than 1PEM because sub-lattices of ions of different masses are affected differently by the passage of electrons through the lattice of the SC. We are hence led to the following outline of a novel strategy for dealing with multicomponent SCs: start with a BSE for the bound states of two fermions, temperature-generalize it via the Matsubara prescription, and, for the kernel of the equation, employ a superpropagator representing exchange of multiple phonons for pairing.

To implement the above program, we need to employ (i) the instantaneous and mean-field approximations (IA, MFA), (ii) a model for fixing the Debye

(1)

ductivity which is a strictly nonrelativistic phenomenon.

ð

*dq*<sup>4</sup> <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

ΔSP

propagator, viz.,

**57**

*<sup>F</sup>* ð Þ*<sup>k</sup>* <sup>≈</sup> <sup>1</sup> *k*2 *<sup>g</sup>*<sup>2</sup> �

Δ<sup>1</sup>�Particle

component q4 of the four-vector q<sup>μ</sup> = (q4, **q**) as follows (e.g., [6]):

�*i<sup>β</sup>* <sup>∑</sup> ∞ *n*¼�∞

constant and ϕ a scalar field.

*DOI: http://dx.doi.org/10.5772/intechopen.84340*

On the theoretical front, a clear explanation of superconductivity evaded the efforts of many theoreticians who had been engaged in the quest. It was not until 46 years had elapsed since Onnes' discovery that John Bardeen, Leon Cooper, and Robert Schrieffer (BCS) [2] were able to provide a generally accepted explanation of it for elemental SCs, for which they were awarded the Nobel Prize in 1972. Indeed, a contributing factor for this lag between theory and experiment was the tragedy of the two world wars in the intervening period.

BCS theory explains superconductivity as due to the formation of Cooper pairs (CPs), each of which is a bound state of two electrons. A well-known explanation for the origin of the attractive interaction causing electrons to be bound together is as follows: the passage of an electron through the ion lattice of an element leaves behind a deformation trail of enhanced positive charges. If a second electron passes through the lattice while it is recovering from the effect of the passage of the first, it will experience an attractive force that may be greater than the Coulomb repulsion between them. Hence, overall, the electron-lattice-electron interaction can be a net *weak* attractive interaction that leads to the formation of CPs. Since the bound state of two particles has lower energy than the state in which they are both free, the former becomes the preferred state leading to the occurrence of superconductivity. In the language of quantum field theory, it is said that the pairing of electrons takes place because they exchange a phonon due to the effect of the ion lattice. BCS theory is hence a weak-coupling theory which works for elemental SCs because the highest Tc for this class of SCs is about 9 K (for Nb). A general perception about this theory is that it is inadequate to explain the occurrence of such high-Tcs as have been observed. This is indeed so if the one-phonon exchange mechanism (1PEM) is considered to be the sole mechanism responsible for pairing. Such a view overlooks the fact that *all* high-Tc SCs are composite materials consisting of sub-lattices of more than one ion species and therefore the theory ought to be generalized to address the situation where CPs may also be formed due to more than 1PEM. Interestingly, as noted below, such a generalization can be carried out based on three theoretical developments that took place in the same decade, 1950s, to which the BCS theory belongs.

#### **1.2 GBCSEs and their conceptual basis**

Two of the three developments alluded to the above are widely known: the invention of the Bethe-Salpeter equation (BSE) in 1951 [3] and, in analogy with quantum field theory, the creation of a finite-temperature field theory (FTFT) by *Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features… DOI: http://dx.doi.org/10.5772/intechopen.84340*

Matsubara [4] in 1955. Not so well-known is the third development that took place in 1954 due to Okubo [5], viz., the introduction of the concept of a superpropagator as the propagator for the non-polynomial field exp (gϕ) where g is a coupling constant and ϕ a scalar field.

BSE is a four-dimensional equation which was invented for the description of bound states of two relativistic particles, hordes of which were being discovered in the late 1940s and early 1950s in cosmic rays and high-energy accelerators around the world. A general perception about the BSE is that it is irrelevant for superconductivity which is a strictly nonrelativistic phenomenon.

FTFT is obtained via an adaption of quantum field theory where time is replaced by temperature as the variable in terms of which evolution of a system is studied. Because temperature is a statistical concept, this is a huge step which converts an equation setup for the bound states of two particles interacting in vacuum into an equation valid in a many-body system. A shortcut known as the Matsubara prescription or recipe for introducing temperature into a BSE is to replace in it the component q4 of the four-vector q<sup>μ</sup> = (q4, **q**) as follows (e.g., [6]):

$$\begin{aligned} q\_4 &= (2n+1)\pi/(-i\beta) \quad \text{for fermions} \\ &= 2\mathbf{n}\pi/(-i\beta) \quad \quad \text{for bosons} \end{aligned} \tag{1}$$

$$\int dq\_4 = \frac{2\pi}{-i\beta} \sum\_{n=-\infty}^{\infty}$$

Okubo's work remained an obscure academic exercise till Salam and collaborators found that non-polynomial theories have inbuilt damping effects. It was therefore hoped that they might provide the means of renormalizing, e.g., the weak interactions. These theories were not pursued any further after the gauge theory of interactions became widely accepted as the correct theory. However, it is relevant in the context of superconductivity to note that the expression for the Feynman superpropagator corresponding to the non-polynomial interaction 1*=*ð Þ 1 þ *gϕ* is [7]:

$$
\Delta\_F^{\rm SP}(k) \approx \frac{1}{k^2 g^2} - \int\_0^\infty dt \frac{e^{-t}(2-t)}{k^2 g^2 + t}. \tag{2}
$$

The comparison of the above with the expression for one-particle Feynman propagator, viz.,

$$
\Delta\_F^{1-\text{Particle}}(k) = \frac{1}{k^2 - i\varepsilon} \tag{3}
$$

reveals that a superpropagator represents a weighted superposition of multiple quanta. The significance of this remark emerges if we recall how the electron-latticeelectron interaction in an elemental SC causes formation of CPs via 1PEM. Appealing to the same picture for a composite SC, it follows that pairing can now also be caused via more than 1PEM because sub-lattices of ions of different masses are affected differently by the passage of electrons through the lattice of the SC. We are hence led to the following outline of a novel strategy for dealing with multicomponent SCs: start with a BSE for the bound states of two fermions, temperature-generalize it via the Matsubara prescription, and, for the kernel of the equation, employ a superpropagator representing exchange of multiple phonons for pairing.

To implement the above program, we need to employ (i) the instantaneous and mean-field approximations (IA, MFA), (ii) a model for fixing the Debye

search in scientific laboratories across the world for other superconducting materials having Tcs higher than that of Hg, the ultimate aim being to find a material which is superconducting at room temperature. While this search led to the discovery of many elements and alloys that are superconductors (SCs), till as late as in the 1980s, the highest Tc of any known SC was about 23 K (for Nb3Sn). A radical change occurred in this situation in 1986 when Bednorz and Müller announced the discovery of an SC (Ba-La-Cu-O system) with *Tc* ≈38 K*,*for which they were awarded the Nobel Prize in 1987. An avalanche of activity followed soon thereafter leading to the discovery of many "high-temperature SCs" such as MgB2 ð Þ *Tc* ≃40 K , Fe-based SCs highest ð Þ *Tc* ≃ 55 K *,* and "cuprates," examples of which are Bi-, Tl-, and Hg-based SCs that contain one or more units of CuO2 as a constituent, the Tcs of these SCs at ambient pressure being about 95, 110, and 138 K, respectively, and for the Hg-based SCs, going up to 164 K at high pressure. The confirmed current record for the highest Tc (203 K; at a pressure of 150 GPa) is held by H3S which was discovered in 2015. It would thus appear that realization of

On the theoretical front, a clear explanation of superconductivity evaded the efforts of many theoreticians who had been engaged in the quest. It was not until 46 years had elapsed since Onnes' discovery that John Bardeen, Leon Cooper, and Robert Schrieffer (BCS) [2] were able to provide a generally accepted explanation of it for elemental SCs, for which they were awarded the Nobel Prize in 1972. Indeed, a contributing factor for this lag between theory and experiment was the

BCS theory explains superconductivity as due to the formation of Cooper pairs (CPs), each of which is a bound state of two electrons. A well-known explanation for the origin of the attractive interaction causing electrons to be bound together is as follows: the passage of an electron through the ion lattice of an element leaves behind a deformation trail of enhanced positive charges. If a second electron passes through the lattice while it is recovering from the effect of the passage of the first, it will experience an attractive force that may be greater than the Coulomb repulsion between them. Hence, overall, the electron-lattice-electron interaction can be a net *weak* attractive interaction that leads to the formation of CPs. Since the bound state of two particles has lower energy than the state in which they are both free, the former becomes the preferred state leading to the occurrence of superconductivity. In the language of quantum field theory, it is said that the pairing of electrons takes place because they exchange a phonon due to the effect of the ion lattice. BCS theory is hence a weak-coupling theory which works for elemental SCs

because the highest Tc for this class of SCs is about 9 K (for Nb). A general perception about this theory is that it is inadequate to explain the occurrence of such high-Tcs as have been observed. This is indeed so if the one-phonon exchange mechanism (1PEM) is considered to be the sole mechanism responsible for pairing.

Such a view overlooks the fact that *all* high-Tc SCs are composite materials consisting of sub-lattices of more than one ion species and therefore the theory ought to be generalized to address the situation where CPs may also be formed due to more than 1PEM. Interestingly, as noted below, such a generalization can be carried out based on three theoretical developments that took place in the same

Two of the three developments alluded to the above are widely known: the invention of the Bethe-Salpeter equation (BSE) in 1951 [3] and, in analogy with quantum field theory, the creation of a finite-temperature field theory (FTFT) by

decade, 1950s, to which the BCS theory belongs.

**1.2 GBCSEs and their conceptual basis**

**56**

the dream of room temperature SCs is near at hand.

*On the Properties of Novel Superconductors*

tragedy of the two world wars in the intervening period.

temperatures of all the ions that may cause pairing, and (iii) the Bogoliubov constraint on the interaction parameters obtained via solutions of the equations that the BSE leads to when multi-phonon exchange mechanisms are operative. An account of these concepts is as follows.

*θCa* ¼ 237 K*, θBi* ¼ 269 K*, θ*Sr ¼ 286 K*:* (6)

We note that (i) θCa equals the Debye temperature of the SC because the layers containing Ca ions do not have any other constituent. (ii) The value of θBi corresponds to Bi being the upper bob of the double pendulum and O the lower bob in the BiO layers; the other value of θ<sup>B</sup> when Bi is the lower bob is 57 K. (iii) The value of θSr also corresponds to Sr. being the upper bob of the double pendulum in the SrO layers; the other value of θSr is 81 K. (iv) The values of θ<sup>O</sup> in the two layers are not noted because they do not play a direct role in the pairing process. (v) Since we have four distinct choices for values of the pair {θBi, θSr}, we need a criterion for choosing one among them, and this is provided by the Bogoliubov constraint

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features…*

In a rigorous study concerned with renormalization of the BCS theory, starting with the assumption of a net attractive interaction between electrons, Bogoliubov found that the interaction parameter λ governing the pairing process must not exceed 0.5, otherwise the system would be unstable; for a detailed discussion, see [9]. Imposition of this constraint on the λs obtained via GBCSEs for high-Tc SCs is crucial. This is so because if one calculates λs for each of the four choices of θs as noted above, one can straightway reject those that lead to any of them that are negative and choose among the rest the ones that lead to values closest to 0.5. The

This chapter is organized as follows. In Section 2.1 is given the parent Bethe-Salpeter equation from which GBCSEs for the Tc and Δs of a multicomponent SC are derived. In Section 2.2, after an outline of the key steps of their derivation, are given the GBCSEs which are constrained by the inequality *EF* ≫ *kθ:* More generally, EF-incorporated GBCSEs, which are not constrained by this inequality, are given in Section 2.3. This is followed up by an account of similar equations, i.e., subject to

eters of an SC. In Section 4 are listed the properties of Bi-2212, the explanation of which is taken up in Section 5. In Section 6 we draw attention to the application of the framework of GBCSEs to a diverse variety of SCs, viz., LCO, the heavy fermion, and the Fe-based SCs. Also included in this section is a brief account of a recent study concerned with the isotope-like effect for Bi-2212. Sections 6 and 7 are devoted, respectively, to the discussion of the key features of our approach and the

After an outline of the key steps of their derivation, we give in this section the GBCSEs that are later employed for the study of various superconducting features of Bi2Sr2CaCu2O8 (Bi-2212). We refer the reader to [10] for a detailed derivation of

The parent equation from which GBCSEs are derived is the following relativistic

BSE [11] for the bound states of fermions 1 and 2 of equal mass (m):

p *,* which is required for calculating j0 and several other param-

and not subject to the said inequality, for the dimensionless construct

values of θs in Eq. (6) have been chosen on this basis.

*DOI: http://dx.doi.org/10.5772/intechopen.84340*

discussed below.

**1.3 Plan of the chapter**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*m=EF*

conclusions following from them.

**2. The framework of GBCSEs**

*y* ¼ ð Þ *kθ=P*<sup>0</sup>

these equations.

**59**

**2.1 The parent BSE**

By IA is meant that we ignore the time of propagation of the quanta, the exchange of which causes the electrons to be bound together. For example, in the momentum-space transform of the one-particle propagator in Eq. (3) where *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> <sup>0</sup> � *<sup>k</sup>*<sup>2</sup> *, k*<sup>2</sup> <sup>0</sup> is equated to zero. This is an essential step for the application of the Matsubara prescription which introduces temperature into the theory and reduces the BSE to a three-dimensional equation which is then subjected to the nonrelativistic approximation appropriate for superconductivity.

Employing MFA in the 1PEM scenario, one approximates the propagator noted in Eq. (3) by a constant, �V (as in [N(0)V], ℏ ¼ *c* ¼ 1), because pairing takes place in a very small region demarcated by *EF* � *<sup>k</sup><sup>θ</sup>* <sup>≤</sup>*p*<sup>2</sup> ð Þ *<sup>=</sup>*2*<sup>m</sup>* <sup>≤</sup>*EF* <sup>þ</sup> *<sup>k</sup><sup>θ</sup> ,* where *EF* denotes the Fermi energy, p2 /2m the energy of an electron, and θ the Debye temperature of the SC; outside of this region, V = 0. Similarly, in the nPEM scenario, in lieu of the expression for the superpropagator in Eq. (2), one employs the expression – [V1 + V2 + … + Vn], where n is the number of ion species that cause the electrons to form a pair, Vi corresponds to the phonon exchanged due to the ith species of ions of which the Debye temperature is θi, and *EF* � *<sup>k</sup>θ<sup>i</sup>* <sup>≤</sup> *<sup>p</sup>*<sup>2</sup> ð Þ *<sup>=</sup>*2*<sup>m</sup>* <sup>≤</sup>*EF* <sup>þ</sup> *<sup>k</sup>θ<sup>i</sup>* is the region of pairing due to these ions, outside of which Vi = 0.

Given the Debye temperature θ of any composite SC, we now need to fix the Debye temperatures of its constituent ion species, θ1, θ2, etc., which in general *must* be different from θ because, as was first pointed out by Born and Karman a long time ago, elastic waves in an anisotropic material travel with different velocities in different directions; for a detailed account of their work, see [8]. We now draw attention to the fact that sublattices of all the high-Tc SCs are composed of layers that contain either one or predominantly two ion species. The assumption that θ is also the Debye temperature of each of these sub-lattices then fixes the Debye temperature of the sub-lattice of layers that contains a single ion species as θ. As concerns fixing θ<sup>1</sup> and θ<sup>2</sup> for the sub-lattice comprising two ion species, we note that the following relation has been frequently used for finding the Debye temperature θ of a binary AxB1-x when the Debye temperatures θ<sup>A</sup> and θ<sup>B</sup> are known:

$$
\theta = \mathfrak{x}\theta\_A + (\mathfrak{1} - \mathfrak{x})\theta\_B. \tag{4}
$$

Since we are faced with the inverse of the above situation, i.e., given θ, we have to fix θ<sup>A</sup> and θB, we now need to supplement Eq. (4) by another equation. We meet this requirement by assuming that the modes of vibration of the A and B ions simulate the weakly coupled modes of vibration of a double pendulum. This is suggested by the polariton effect where coupling between the phonon and the photon modes leads to two new modes, different from the original modes of either of them. With A as the upper bob in the double pendulum, we have our second equation as [10].

$$\frac{\theta\_A}{\theta\_B} = \left[ \frac{1 + \sqrt{m\_B/(m\_A + m\_B)}}{1 - \sqrt{m\_B/(m\_A + m\_B)}} \right]^{1/2},\tag{5}$$

where mA and mB are the atomic masses of the ions.

Given θ (Bi2Sr2CaCu2O8) = 237 K, mBi = 208.98, mSr = 87.62, and mO = 15.999 (amu), the application of Eqs. (4) and (5) leads to

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features… DOI: http://dx.doi.org/10.5772/intechopen.84340*

$$
\theta\_{\text{Cat}} = \text{237 K}, \theta\_{\text{Bi}} = \text{269 K}, \theta\_{\text{Sr}} = \text{286 K}. \tag{6}
$$

We note that (i) θCa equals the Debye temperature of the SC because the layers containing Ca ions do not have any other constituent. (ii) The value of θBi corresponds to Bi being the upper bob of the double pendulum and O the lower bob in the BiO layers; the other value of θ<sup>B</sup> when Bi is the lower bob is 57 K. (iii) The value of θSr also corresponds to Sr. being the upper bob of the double pendulum in the SrO layers; the other value of θSr is 81 K. (iv) The values of θ<sup>O</sup> in the two layers are not noted because they do not play a direct role in the pairing process. (v) Since we have four distinct choices for values of the pair {θBi, θSr}, we need a criterion for choosing one among them, and this is provided by the Bogoliubov constraint discussed below.

In a rigorous study concerned with renormalization of the BCS theory, starting with the assumption of a net attractive interaction between electrons, Bogoliubov found that the interaction parameter λ governing the pairing process must not exceed 0.5, otherwise the system would be unstable; for a detailed discussion, see [9]. Imposition of this constraint on the λs obtained via GBCSEs for high-Tc SCs is crucial. This is so because if one calculates λs for each of the four choices of θs as noted above, one can straightway reject those that lead to any of them that are negative and choose among the rest the ones that lead to values closest to 0.5. The values of θs in Eq. (6) have been chosen on this basis.

#### **1.3 Plan of the chapter**

temperatures of all the ions that may cause pairing, and (iii) the Bogoliubov constraint on the interaction parameters obtained via solutions of the equations that the BSE leads to when multi-phonon exchange mechanisms are operative. An account

By IA is meant that we ignore the time of propagation of the quanta, the exchange of which causes the electrons to be bound together. For example, in the momentum-space transform of the one-particle propagator in Eq. (3) where

the Matsubara prescription which introduces temperature into the theory and reduces the BSE to a three-dimensional equation which is then subjected to the

Employing MFA in the 1PEM scenario, one approximates the propagator noted in Eq. (3) by a constant, �V (as in [N(0)V], ℏ ¼ *c* ¼ 1), because pairing takes place in a very small region demarcated by *EF* � *<sup>k</sup><sup>θ</sup>* <sup>≤</sup>*p*<sup>2</sup> ð Þ *<sup>=</sup>*2*<sup>m</sup>* <sup>≤</sup>*EF* <sup>þ</sup> *<sup>k</sup><sup>θ</sup> ,*

Debye temperature of the SC; outside of this region, V = 0. Similarly, in the nPEM scenario, in lieu of the expression for the superpropagator in Eq. (2), one employs the expression – [V1 + V2 + … + Vn], where n is the number of ion species that cause the electrons to form a pair, Vi corresponds to the phonon exchanged due to the ith species of ions of which the Debye temperature is θi, and *EF* � *<sup>k</sup>θ<sup>i</sup>* <sup>≤</sup> *<sup>p</sup>*<sup>2</sup> ð Þ *<sup>=</sup>*2*<sup>m</sup>* <sup>≤</sup>*EF* <sup>þ</sup> *<sup>k</sup>θ<sup>i</sup>* is the region of pairing due to these ions, outside

Given the Debye temperature θ of any composite SC, we now need to fix the Debye temperatures of its constituent ion species, θ1, θ2, etc., which in general *must* be different from θ because, as was first pointed out by Born and Karman a long time ago, elastic waves in an anisotropic material travel with different velocities in different directions; for a detailed account of their work, see [8]. We now draw attention to the fact that sublattices of all the high-Tc SCs are composed of layers that contain either one or predominantly two ion species. The assumption that θ is also the Debye temperature of each of these sub-lattices then fixes the Debye temperature of the sub-lattice of layers that contains a single ion species as θ. As concerns fixing θ<sup>1</sup> and θ<sup>2</sup> for the sub-lattice comprising two ion species, we note that the following relation has been frequently used for finding the Debye temperature θ

Since we are faced with the inverse of the above situation, i.e., given θ, we have to fix θ<sup>A</sup> and θB, we now need to supplement Eq. (4) by another equation. We meet this requirement by assuming that the modes of vibration of the A and B ions simulate the weakly coupled modes of vibration of a double pendulum. This is suggested by the polariton effect where coupling between the phonon and the photon modes leads to two new modes, different from the original modes of either of them. With A as the upper bob in the double pendulum, we have our second

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *mB=*ð Þ *mA* þ *mB*

Given θ (Bi2Sr2CaCu2O8) = 237 K, mBi = 208.98, mSr = 87.62, and mO = 15.999

" #<sup>1</sup>*=*<sup>2</sup>

p

p

*mB=*ð Þ *mA* þ *mB*

of a binary AxB1-x when the Debye temperatures θ<sup>A</sup> and θ<sup>B</sup> are known:

*θA θB*

(amu), the application of Eqs. (4) and (5) leads to

where mA and mB are the atomic masses of the ions.

nonrelativistic approximation appropriate for superconductivity.

<sup>0</sup> is equated to zero. This is an essential step for the application of

/2m the energy of an electron, and θ the

*θ* ¼ *xθ<sup>A</sup>* þ ð Þ 1 � *x θB:* (4)

*,* (5)

of these concepts is as follows.

*On the Properties of Novel Superconductors*

where *EF* denotes the Fermi energy, p2

*<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

<sup>0</sup> � *<sup>k</sup>*<sup>2</sup> *, k*<sup>2</sup>

of which Vi = 0.

equation as [10].

**58**

This chapter is organized as follows. In Section 2.1 is given the parent Bethe-Salpeter equation from which GBCSEs for the Tc and Δs of a multicomponent SC are derived. In Section 2.2, after an outline of the key steps of their derivation, are given the GBCSEs which are constrained by the inequality *EF* ≫ *kθ:* More generally, EF-incorporated GBCSEs, which are not constrained by this inequality, are given in Section 2.3. This is followed up by an account of similar equations, i.e., subject to and not subject to the said inequality, for the dimensionless construct *y* ¼ ð Þ *kθ=P*<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*m=EF* p *,* which is required for calculating j0 and several other parameters of an SC. In Section 4 are listed the properties of Bi-2212, the explanation of which is taken up in Section 5. In Section 6 we draw attention to the application of the framework of GBCSEs to a diverse variety of SCs, viz., LCO, the heavy fermion, and the Fe-based SCs. Also included in this section is a brief account of a recent study concerned with the isotope-like effect for Bi-2212. Sections 6 and 7 are devoted, respectively, to the discussion of the key features of our approach and the conclusions following from them.

#### **2. The framework of GBCSEs**

After an outline of the key steps of their derivation, we give in this section the GBCSEs that are later employed for the study of various superconducting features of Bi2Sr2CaCu2O8 (Bi-2212). We refer the reader to [10] for a detailed derivation of these equations.

#### **2.1 The parent BSE**

The parent equation from which GBCSEs are derived is the following relativistic BSE [11] for the bound states of fermions 1 and 2 of equal mass (m):

$$\begin{split} \, \nu \left( p\_{\mu} \right) &= (-2\pi i)^{-1} \int d^{4}q\_{\mu} \frac{1}{\gamma\_{\mu}^{(1)} P\_{\mu}/2 + \gamma\_{\mu}^{(1)} q\_{\mu} - m + i\varepsilon} \\ &\propto \frac{1}{\gamma\_{\mu}^{(2)} P\_{\mu}/2 + \gamma\_{\mu}^{(2)} q\_{\mu} - m + i\varepsilon} I \Big( q\_{\mu} - p\_{\mu} \Big) \nu \left( q\_{\mu} \right), \end{split} \tag{7}$$

where ψ is the BS amplitude for the formation of the bound state of the two particles, P<sup>μ</sup> the total four-momentum of their center of mass, q<sup>μ</sup> their relative fourmomentum, *γ*ð Þ <sup>1</sup>*;*<sup>2</sup> *<sup>μ</sup>* are their Dirac matrices, and I(q<sup>μ</sup> – pμ) is the kernel of the equation that causes the particles to be bound together; the metric employed is time-preferred: aμb<sup>μ</sup> = a4b4 � **a.b**.

#### **2.2 GBCSEs corresponding to multiphonon exchanges for pairing**

We now outline how GBCSEs are derived when CPs in an SC are formed due to multi-phonon exchange mechanisms subject to the following constraint:

$$E\_F >> k\theta,\tag{8}$$

highest *Tc* are due to a three-phonon exchange mechanism (3PEM) (reminder:

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features…*

2*kθ*<sup>1</sup> ∣*W*2∣ � �

þ *λ*<sup>2</sup> ln 1 þ

ð*<sup>θ</sup>*2*=*2*kTc* 0

þ *λ*<sup>2</sup> ln 1 þ

2*kθ*<sup>2</sup> ∣*W*3∣ � �

þ *λ*<sup>3</sup>

*<sup>F</sup> <sup>=</sup>*4*π*<sup>2</sup> ð Þ <sup>ℏ</sup> <sup>¼</sup> *<sup>c</sup>* <sup>¼</sup> <sup>1</sup> *:*

*dx* tanh ð Þ *<sup>x</sup> x*

*E*1*=*<sup>2</sup>

∣*W*1∣ ¼ 2*kθ=*½ � exp 1ð Þ� *=λ* 1 *:*

Δ<sup>0</sup> ¼ *kθ=*sinh 1ð Þ *=λ ,*

we observe that j j¼ *W*<sup>1</sup> Δ<sup>0</sup> ¼ 2*kθ* exp ð Þ �1*=λ ,* when *λ* ! 0*:* This per se suggests that one may employ the equation for∣*W*1∣ in lieu of the equation for *Δ0*. Based on the values of ∣W1∣ and Δ<sup>0</sup> obtained via a detailed study of six elemental SCs [10] by employing the empirical values of their Debye temperatures, it has been shown that ∣W1∣ is a viable alternative to Δ0*:* We are thus led to assume that each of the three Ws following from Eq. (11) is to be identified with a gap of the SC and that ∣W3∣>∣W2∣>∣W1∣*,* where ∣W2∣ corresponds to the solution of the equation when one

It has been known for a long time that Tcs of some SCs—SrTiO3 being a prime example—depend in a marked manner on their electron concentration ns which is related to EF. Recent studies—both experimental [12–14] and theoretical [15–17] too suggest that *low* values of *EF* play a fundamental role in determining the properties of high-Tc SCs. This makes it natural to seek equations for the Tcs and *Δ*s of these SCs that incorporate EF as a variable and are therefore not constrained by inequality (8). Such equations are in fact an integral part of the BCS theory. Their application to dilute SrTiO3 by Eagles [18] perhaps marks the beginning of what is now known as BCS-BEC crossover physics. It seems to us that the focus of crossover physics has, by and large, been different from that of the studies of high-Tc SCs. Beginning with Eq. (7) and carrying out steps (1–8) noted above, without

<sup>2</sup> *<sup>a</sup>*<sup>2</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ <sup>3</sup>

4

<sup>2</sup> *<sup>b</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ <sup>3</sup>

*<sup>a</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ � �<sup>1</sup>*=*<sup>3</sup>

4

*<sup>b</sup>*<sup>4</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ � �<sup>1</sup>*=*<sup>3</sup>

(13)

(14)

imposing inequality (8), the μ-incorporated GBCSEs corresponding to

<sup>2</sup> *<sup>b</sup>*<sup>2</sup> *<sup>μ</sup>*<sup>0</sup> ð Þþ *<sup>λ</sup>*<sup>3</sup>

Recalling the BCS equation for the gap of an elemental SC at *T* = 0, i.e.,

Remark: If *λ<sup>2</sup>* = 0 in Eq. (10) and *λ<sup>2</sup>* = *λ*<sup>3</sup> = 0 in both Eqs. (11) and (12), then we have a one-phonon exchange mechanism (1PEM) in operation. In this case Eq. (12) becomes identical with the BCS equation for the *Tc* of an elemental SC, whereas

2*kθ*<sup>2</sup> ∣*W*2∣

þ *λ*<sup>3</sup> ln 1 þ

ð*<sup>θ</sup>*3*=*2*kTc* 0

� � (10)

2*kθ*<sup>3</sup> ∣*W*3∣ � � (11)

*dx* tanh ð Þ *<sup>x</sup>*

*<sup>x</sup> ,* (12)

1 ¼ *λ*<sup>1</sup> ln 1 þ

2*kθ*<sup>1</sup> ∣*W*3∣ � �

þ *λ*<sup>2</sup>

of the λs is zero and ∣W1∣ to when two of them are zero.

1 ¼ *λ*<sup>1</sup> ln 1 þ

*DOI: http://dx.doi.org/10.5772/intechopen.84340*

*dx* tanh ð Þ *<sup>x</sup> x*

where *<sup>λ</sup><sup>i</sup>* <sup>¼</sup> *<sup>N</sup>*ð Þ <sup>0</sup> *Vi* ½ �*, N*ð Þ¼ <sup>0</sup> ð Þ <sup>2</sup>*<sup>m</sup>* <sup>3</sup>*=*<sup>2</sup>

EF >> kθ)

1 ¼ *λ*<sup>1</sup>

ð*<sup>θ</sup>*1*=*2*kTc* 0

Eq. (10), or Eq. (11), leads to

**2.3 EF-incorporated GBCSEs**

Eqs. (10)–(12) are as follows [10]:

*λ*1

**61**

*λ*1

<sup>2</sup> *<sup>b</sup>*<sup>1</sup> *<sup>μ</sup>*<sup>0</sup> ð Þþ *<sup>λ</sup>*<sup>2</sup>

<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>μ</sup>*<sup>0</sup> ð Þþ *<sup>λ</sup>*<sup>2</sup>

where θ is the Debye temperature of the SC. Beginning with Eq. (7), we obtain GBCSEs by:


$$I(\mathbf{q} - \mathbf{p}) = \sum\_{i=1}^{n} \frac{-V\_i}{\left(2\pi\right)^3} \left( \text{for } E\_F - k\theta\_i \le \frac{\mathbf{p}^2}{2m}, \frac{\mathbf{q}^2}{2m} \le E\_F + k\theta\_i \right) \tag{9}$$
 
$$= \mathbf{0} \text{ (otherwise)},$$

where *n* is the number of ion species due to which pairs are formed and θ<sup>i</sup> is the Debye temperature of the ith species of ions

8.Calculating θi via the Debye temperature θ of the SC and Eqs. (4) and (5) given above

With their application to Bi-2212 in mind, given below are the equations that the above steps lead to in the scenario in which the smaller of the two gaps of an SC is due to a two-phonon exchange mechanism (2PEM) and its larger gap and the

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features… DOI: http://dx.doi.org/10.5772/intechopen.84340*

highest *Tc* are due to a three-phonon exchange mechanism (3PEM) (reminder: EF >> kθ)

$$\mathbf{1} = \lambda\_1 \ln\left[\mathbf{1} + \frac{2k\theta\_1}{|W\_2|}\right] + \lambda\_2 \ln\left[\mathbf{1} + \frac{2k\theta\_2}{|W\_2|}\right] \tag{10}$$

$$\mathbf{1} = \lambda\_1 \ln\left[\mathbf{1} + \frac{2k\theta\_1}{|W\_3|}\right] + \lambda\_2 \ln\left[\mathbf{1} + \frac{2k\theta\_2}{|W\_3|}\right] + \lambda\_3 \ln\left[\mathbf{1} + \frac{2k\theta\_3}{|W\_3|}\right] \tag{11}$$

$$\mathbf{1} = \lambda\_1 \int\_0^{\theta\_1/2kT\_\varepsilon} d\mathbf{x} \, \frac{\tanh\left(\mathbf{x}\right)}{\mathbf{x}} + \lambda\_2 \int\_0^{\theta\_2/2kT\_\varepsilon} d\mathbf{x} \, \frac{\tanh\left(\mathbf{x}\right)}{\mathbf{x}} + \lambda\_3 \int\_0^{\theta\_3/2kT\_\varepsilon} d\mathbf{x} \, \frac{\tanh\left(\mathbf{x}\right)}{\mathbf{x}},\tag{12}$$

where *<sup>λ</sup><sup>i</sup>* <sup>¼</sup> *<sup>N</sup>*ð Þ <sup>0</sup> *Vi* ½ �*, N*ð Þ¼ <sup>0</sup> ð Þ <sup>2</sup>*<sup>m</sup>* <sup>3</sup>*=*<sup>2</sup> *E*1*=*<sup>2</sup> *<sup>F</sup> <sup>=</sup>*4*π*<sup>2</sup> ð Þ <sup>ℏ</sup> <sup>¼</sup> *<sup>c</sup>* <sup>¼</sup> <sup>1</sup> *:*

Remark: If *λ<sup>2</sup>* = 0 in Eq. (10) and *λ<sup>2</sup>* = *λ*<sup>3</sup> = 0 in both Eqs. (11) and (12), then we have a one-phonon exchange mechanism (1PEM) in operation. In this case Eq. (12) becomes identical with the BCS equation for the *Tc* of an elemental SC, whereas Eq. (10), or Eq. (11), leads to

$$|W\_1| = 2k\theta/[\exp\left(\mathbf{1}/\lambda\right) - \mathbf{1}].$$

Recalling the BCS equation for the gap of an elemental SC at *T* = 0, i.e.,

$$
\Delta\_0 = k\theta / \sinh\left(\mathbf{1}/\lambda\right),
$$

we observe that j j¼ *W*<sup>1</sup> Δ<sup>0</sup> ¼ 2*kθ* exp ð Þ �1*=λ ,* when *λ* ! 0*:* This per se suggests that one may employ the equation for∣*W*1∣ in lieu of the equation for *Δ0*. Based on the values of ∣W1∣ and Δ<sup>0</sup> obtained via a detailed study of six elemental SCs [10] by employing the empirical values of their Debye temperatures, it has been shown that ∣W1∣ is a viable alternative to Δ0*:* We are thus led to assume that each of the three Ws following from Eq. (11) is to be identified with a gap of the SC and that ∣W3∣>∣W2∣>∣W1∣*,* where ∣W2∣ corresponds to the solution of the equation when one of the λs is zero and ∣W1∣ to when two of them are zero.

### **2.3 EF-incorporated GBCSEs**

It has been known for a long time that Tcs of some SCs—SrTiO3 being a prime example—depend in a marked manner on their electron concentration ns which is related to EF. Recent studies—both experimental [12–14] and theoretical [15–17] too suggest that *low* values of *EF* play a fundamental role in determining the properties of high-Tc SCs. This makes it natural to seek equations for the Tcs and *Δ*s of these SCs that incorporate EF as a variable and are therefore not constrained by inequality (8). Such equations are in fact an integral part of the BCS theory. Their application to dilute SrTiO3 by Eagles [18] perhaps marks the beginning of what is now known as BCS-BEC crossover physics. It seems to us that the focus of crossover physics has, by and large, been different from that of the studies of high-Tc SCs.

Beginning with Eq. (7) and carrying out steps (1–8) noted above, without imposing inequality (8), the μ-incorporated GBCSEs corresponding to Eqs. (10)–(12) are as follows [10]:

$$\frac{\lambda\_1}{2}a\_1(\mu\_0) + \frac{\lambda\_2}{2}a\_2(\mu\_0) = \left[\frac{3}{4}a\_3(\mu\_0)\right]^{1/3} \tag{13}$$

$$
\frac{\lambda\_1}{2}b\_1(\mu\_0) + \frac{\lambda\_2}{2}b\_2(\mu\_0) + \frac{\lambda\_3}{2}b\_3(\mu\_0) = \left[\frac{3}{4}b\_4(\mu\_0)\right]^{1/3} \tag{14}
$$

*ψ p<sup>μ</sup>* � �

*On the Properties of Novel Superconductors*

time-preferred: aμb<sup>μ</sup> = a4b4 � **a.b**.

1. Employing IA: *I q<sup>μ</sup>* � *p<sup>μ</sup>*

ð Þ1 <sup>4</sup> *γ* ð Þ2 4

4.Putting *E* ¼ 2*EF* þ *W*

previous section

above

**60**

*<sup>I</sup> <sup>q</sup>* � *<sup>p</sup>* � � <sup>¼</sup> <sup>∑</sup>

*n i*¼1

the Debye temperature of the ith species of ions

�*Vi*

¼ 0 otherwise ð Þ*,*

matrices *γ*

¼ �ð Þ <sup>2</sup>*π<sup>i</sup>* �<sup>1</sup> <sup>Ð</sup>

*γ* ð Þ2 *<sup>μ</sup> Pμ=*2 þ *γ*

where θ is the Debye temperature of the SC. Beginning with Eq. (7), we obtain GBCSEs by:

� �

<sup>x</sup> <sup>1</sup>

*d*4 *qμ*

ð Þ2

momentum, *γ*ð Þ <sup>1</sup>*;*<sup>2</sup> *<sup>μ</sup>* are their Dirac matrices, and I(q<sup>μ</sup> – pμ) is the kernel of the equation that causes the particles to be bound together; the metric employed is

**2.2 GBCSEs corresponding to multiphonon exchanges for pairing**

multi-phonon exchange mechanisms subject to the following constraint:

<sup>¼</sup> *<sup>I</sup> <sup>q</sup>* � *<sup>p</sup>* � �

3.Working in the rest frame of the two particles: *P<sup>μ</sup>* ¼ ð Þ *E;* 0

2. Carrying out spin reduction of the equation by multiplying it with Dirac

5. Temperature-generalizing the equation via the Matsubara recipe given in the

6.Assuming that the signature of W changes on crossing the Fermi surface

7. Choosing for the kernel a generalized form of the model BCS interaction:

ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup> for *EF* � *<sup>k</sup>θ<sup>i</sup>* <sup>≤</sup> *<sup>p</sup>*<sup>2</sup>

where *n* is the number of ion species due to which pairs are formed and θ<sup>i</sup> is

8.Calculating θi via the Debye temperature θ of the SC and Eqs. (4) and (5) given

With their application to Bi-2212 in mind, given below are the equations that the above steps lead to in the scenario in which the smaller of the two gaps of an SC is due to a two-phonon exchange mechanism (2PEM) and its larger gap and the

<sup>2</sup>*<sup>m</sup> ; q*2 2*m*

� �

≤*EF* þ *kθ<sup>i</sup>*

(9)

*γ* ð Þ1 *<sup>μ</sup> Pμ=*2 þ *γ*

*<sup>μ</sup> q<sup>μ</sup>* � *m* þ *iε*

We now outline how GBCSEs are derived when CPs in an SC are formed due to

where ψ is the BS amplitude for the formation of the bound state of the two particles, P<sup>μ</sup> the total four-momentum of their center of mass, q<sup>μ</sup> their relative four-

1

ð Þ1

*I q<sup>μ</sup>* � *p<sup>μ</sup>* � �

*<sup>μ</sup> q<sup>μ</sup>* � *m* þ *iε*

*EF*>>*kθ,* (8)

*ψ q<sup>μ</sup>* � � *,* (7)

$$
\frac{\lambda\_1}{2}c\_1(\mu\_1) + \frac{\lambda\_2}{2}c\_2(\mu\_1) + \frac{\lambda\_3}{2}c\_3(\mu\_1) = \left[\frac{3}{4}c\_4(\mu\_1)\right]^{1/3},\tag{15}
$$

superconducting electrons ns at T = 0, and hence for j0 in terms of y and the values of the following other parameters of the SC: θ, the electronic specific heat constant

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features…*

volume of the SC.

where

[19, 20] that

and hence

*j* <sup>0</sup>ð Þ¼ *EF*

*<sup>A</sup>*<sup>3</sup> ffi <sup>1</sup>*:*584*x*10�<sup>6</sup> cm K�1*=*<sup>3</sup>

*<sup>A</sup>*<sup>5</sup> ffi <sup>6</sup>*:*146*x*10�<sup>4</sup> CeV�4*=*<sup>3</sup>

**63**

*<sup>A</sup>*<sup>1</sup> ffi <sup>3</sup>*:*305*x*10�<sup>10</sup> eV�1*=*<sup>3</sup>

been determined by solving Eqs. (10)–(12):

*DOI: http://dx.doi.org/10.5772/intechopen.84340*

equation for y in the 2PEM (1PEM) scenario.

*t r*ð Þ¼ *; <sup>y</sup> r y* ln *r y*

*r y* � 1 � �

*<sup>s</sup>* � *<sup>m</sup>*<sup>∗</sup> *<sup>=</sup>me* <sup>¼</sup> *<sup>A</sup>*<sup>1</sup>

*P*0ð Þ¼ *EF A*<sup>3</sup>

*v*0ð Þ¼ *EF A*<sup>4</sup>

*<sup>e</sup>* <sup>∗</sup> *<sup>v</sup>*0ð Þ¼ *EF <sup>A</sup>*<sup>5</sup>

cm2K4*<sup>=</sup>*<sup>3</sup>

*ns*ð Þ *EF* 2

> K<sup>1</sup>*=*<sup>3</sup> s�<sup>1</sup>*:*

where *e* is the electronic charge, and

**2.5** *EF***-dependent equation for** *y* ¼ ð Þ *kθ=P***<sup>0</sup>**

*ns*ð Þ¼ *EF A*<sup>2</sup> *γ=vg*

*θ y*

*θ y*

Note that equating one (two) of the λs in Eq. (16) to zero, we obtain the

In terms of y and the other parameters mentioned above, it has been shown in

<sup>γ</sup>, *<sup>s</sup>* � *<sup>m</sup>*<sup>∗</sup> *<sup>=</sup>me,* v0, ns, and vg, where me is the free electron mass and vg the gram-atomic

As concerns y, we recall that in the original BCS theory, the Hamiltonian was restricted at the outset to comprise terms corresponding to pairs having zero centerof-mass momentum. The BSE-based approach enables one to go beyond this restriction by setting *P<sup>μ</sup>* ¼ ð Þ *E; P ,* rather than ð Þ *E;* 0 *,* where **P** is the threemomentum of CPs in the lab frame. Beginning with Eq. (7) and carrying out essentially the same steps as for the *P<sup>μ</sup>* ¼ ð Þ *E;* 0 case, we now obtain in the 3PEM scenario the following equation for y [10], which can be solved after the λs have

1 ¼ *λ*<sup>1</sup> *t r*ð Þþ <sup>1</sup>*; y λ*<sup>2</sup> *t r*ð Þþ <sup>2</sup>*; y λ*<sup>3</sup> *t r*ð Þ <sup>3</sup>*; y ,* (16)

þ ln ð Þ *r y* � 1 � �*,*r*<sup>i</sup>* <sup>¼</sup> *<sup>θ</sup>i=θ:*

> *γ=vg* � �<sup>2</sup>*=*<sup>3</sup> *E*<sup>1</sup>*=*<sup>3</sup> *F*

*γ=vg* � �<sup>1</sup>*=*<sup>3</sup> *E*<sup>2</sup>*=*<sup>3</sup> *F*

1

*E*<sup>1</sup>*=*<sup>3</sup> *F*

*E*<sup>2</sup>*=*<sup>3</sup>

K2 *,*

*,* and

*γ=vg* � �<sup>1</sup>*=*<sup>3</sup>

> *θ y γ=vg* � �<sup>2</sup>*=*<sup>3</sup>

*, A*<sup>4</sup> ffi <sup>1</sup>*:*406*x*10<sup>8</sup> eV2*<sup>=</sup>*<sup>3</sup> sec �1K�5*=*<sup>3</sup>

If the Tc and Δs of an SC are studied via Eqs. (10)–(12), which are valid when EF >> kθ, then it is appropriate to employ Eq. (16) for y because it too is valid when the same inequality is satisfied. If it turns out that we need to employ the μ-incorporated Eqs. (13)–(15) in lieu of Eqs. (10)–(12), then consistency demands that we employ

for y, in lieu of Eq. (16), an equation that too is explicitly μ-dependent.

*, A*<sup>2</sup> ffi <sup>2</sup>*:*729*x*10<sup>7</sup> eV�<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*m=E<sup>F</sup>* p

� �*EF* (18)

*<sup>F</sup> , e* <sup>∗</sup> ð Þ <sup>¼</sup> <sup>2</sup>*<sup>e</sup>* (21)

(17)

(19)

(20)

where μ<sup>0</sup> (μ1) is the chemical potential at T = 0 (T = Tc) and μ<sup>0</sup> has been used interchangeably with EF,

*<sup>a</sup>*<sup>1</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>ð</sup>*kθ*<sup>1</sup> �*kθ*<sup>1</sup> *dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p ∣*ξ*∣ þ ∣*W*2∣*=*2 " #*, a*<sup>2</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>ð</sup>*kθ*<sup>2</sup> �*kθ*<sup>2</sup> *dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p ∣*ξ*∣ þ ∣*W*2∣*=*2 " # *<sup>a</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>4</sup> <sup>3</sup> ð Þ *<sup>μ</sup>*<sup>0</sup> � *<sup>k</sup>θ<sup>m</sup>* <sup>3</sup>*=*<sup>2</sup> <sup>þ</sup> ð*kθ<sup>m</sup>* �*kθ<sup>m</sup> dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> <sup>p</sup> <sup>ð</sup><sup>1</sup> � *<sup>ξ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>W</sup>*<sup>2</sup> 2 q 2 6 4 3 7 5 *<sup>b</sup>*<sup>1</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>ð</sup>*kθ*<sup>1</sup> �*kθ*<sup>1</sup> *dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p ∣*ξ*∣ þ ∣*W*3∣*=*2 " #*, b*<sup>2</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>ð</sup>*kθ*<sup>2</sup> �*kθ*<sup>2</sup> *dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p ∣*ξ*∣ þ ∣*W*3∣*=*2 " # *<sup>b</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>ð</sup>*<sup>k</sup>θ*<sup>3</sup> �*kθ*<sup>3</sup> *dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p ∣*ξ*∣ þ ∣*W*3∣*=*2 " # *<sup>b</sup>*<sup>4</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>4</sup> <sup>3</sup> ð Þ *<sup>μ</sup>*<sup>0</sup> � *<sup>k</sup>θ*<sup>2</sup> <sup>3</sup>*=*<sup>2</sup> <sup>þ</sup> ð*<sup>k</sup>θ<sup>m</sup>* �*kθ<sup>m</sup> dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> <sup>p</sup> <sup>1</sup> � *<sup>ξ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>W</sup>*<sup>2</sup> 3 q 0 B@ 3 7 5 2 6 4 *<sup>c</sup>*<sup>1</sup> *<sup>μ</sup>*<sup>1</sup> ð Þ¼ *Re* <sup>ð</sup>*<sup>θ</sup>*1*=*2*Tc*<sup>3</sup> �*θ*1*=*2*Tc*<sup>3</sup> *dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup> 1 tanh ð Þ *x x* " #*, <sup>c</sup>*<sup>2</sup> *<sup>μ</sup>*<sup>1</sup> ð Þ¼ *Re* <sup>ð</sup>*<sup>θ</sup>*2*=*2*Tc*<sup>3</sup> �*θ*2*=*2*Tc*<sup>3</sup> *dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup> 1 tanh ð Þ *x x* " # *<sup>c</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>1</sup> ð Þ¼ *Re* <sup>ð</sup>*<sup>θ</sup>*3*=*2*Tc*<sup>3</sup> �*θ*3*=*2*Tc*<sup>3</sup> *dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup> 1 tanh ð Þ *x x* " # *c*<sup>4</sup> *μ*<sup>1</sup> ð Þ¼ *Re* 2*kTc*<sup>3</sup> ð*<sup>θ</sup>m=*2*Tc*<sup>3</sup> �*θm=*2*Tc*<sup>3</sup> *dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup> <sup>1</sup>f g 1 � tanh ð Þ *x* " #*,*

θ<sup>m</sup> is the Debye temperature that has the highest value among all the Debye temperatures being used (when *λ*<sup>2</sup> 6¼ 0*, θ<sup>m</sup>* ¼ *θ*2*;* when *λ*<sup>2</sup> ¼ 0*, θ<sup>m</sup>* ¼ *θ*3*;* when *λ*<sup>2</sup> ¼ *λ*<sup>3</sup> ¼ 0*, θ<sup>m</sup>* ¼ *θ*1), and the operator *Re* ensures that the integrals yield real values even when expressions under the radical signs are negative.

#### **2.4 Equation for** *y* ¼ ð Þ *kθ=P***<sup>0</sup>** ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi **2***m=E<sup>F</sup>* <sup>p</sup> ð Þ *EF*>>*k<sup>θ</sup>*

If it turns out that we need to employ the *μ*-incorporated Eqs. (13)–(15) to deal with high-Tc SCs, rather than Eqs. (10)–(12), as will be shown to be the case, then there will be a need for at least one more equation for an additional property of the SC, besides equations for one of its Tc and Δs, in order to fix the values of μs and *λ*s that lead to various empirical features of the SC. We choose the critical current density j0 of the SC at T = 0 to meet this need.

The BSE-based approach enables one to derive an equation for j0 of an SC by employing the same values of *θ*s and *λ*s that determine a given value of its Tc and different values of its Δs. This comes about because one can now obtain expressions for the effective mass m\* of an electron, critical velocity v0 at T = 0, and the number of

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features… DOI: http://dx.doi.org/10.5772/intechopen.84340*

superconducting electrons ns at T = 0, and hence for j0 in terms of y and the values of the following other parameters of the SC: θ, the electronic specific heat constant <sup>γ</sup>, *<sup>s</sup>* � *<sup>m</sup>*<sup>∗</sup> *<sup>=</sup>me,* v0, ns, and vg, where me is the free electron mass and vg the gram-atomic volume of the SC.

As concerns y, we recall that in the original BCS theory, the Hamiltonian was restricted at the outset to comprise terms corresponding to pairs having zero centerof-mass momentum. The BSE-based approach enables one to go beyond this restriction by setting *P<sup>μ</sup>* ¼ ð Þ *E; P ,* rather than ð Þ *E;* 0 *,* where **P** is the threemomentum of CPs in the lab frame. Beginning with Eq. (7) and carrying out essentially the same steps as for the *P<sup>μ</sup>* ¼ ð Þ *E;* 0 case, we now obtain in the 3PEM scenario the following equation for y [10], which can be solved after the λs have been determined by solving Eqs. (10)–(12):

$$\mathbf{1} = \lambda\_1 \mathbf{t}(r\_1, \mathbf{y}) + \lambda\_2 \mathbf{t}(r\_2, \mathbf{y}) + \lambda\_3 \mathbf{t}(r\_3, \mathbf{y}), \tag{16}$$

where

*λ*1 2

*On the Properties of Novel Superconductors*

ð*kθ*<sup>1</sup> �*kθ*<sup>1</sup> *dξ*

ð*kθ*<sup>1</sup> �*kθ*<sup>1</sup> *dξ*

ð*<sup>k</sup>θ*<sup>3</sup> �*kθ*<sup>3</sup> *dξ*

2 6 4

2 6 4

interchangeably with EF,

*a*<sup>1</sup> *μ*<sup>0</sup> ð Þ¼ *Re*

*<sup>a</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>4</sup>

*b*<sup>1</sup> *μ*<sup>0</sup> ð Þ¼ *Re*

*b*<sup>3</sup> *μ*<sup>0</sup> ð Þ¼ *Re*

*<sup>b</sup>*<sup>4</sup> *<sup>μ</sup>*<sup>0</sup> ð Þ¼ *Re* <sup>4</sup>

*c*<sup>1</sup> *μ*<sup>1</sup> ð Þ¼ *Re*

*c*<sup>2</sup> *μ*<sup>1</sup> ð Þ¼ *Re*

*c*<sup>3</sup> *μ*<sup>1</sup> ð Þ¼ *Re*

*c*<sup>4</sup> *μ*<sup>1</sup> ð Þ¼ *Re* 2*kTc*<sup>3</sup>

**2.4 Equation for** *y* ¼ ð Þ *kθ=P***<sup>0</sup>**

**62**

*<sup>c</sup>*<sup>1</sup> *<sup>μ</sup>*<sup>1</sup> ð Þþ *<sup>λ</sup>*<sup>2</sup>

2

" #

<sup>3</sup> ð Þ *<sup>μ</sup>*<sup>0</sup> � *<sup>k</sup>θ<sup>m</sup>* <sup>3</sup>*=*<sup>2</sup> <sup>þ</sup>

" #

" #

<sup>3</sup> ð Þ *<sup>μ</sup>*<sup>0</sup> � *<sup>k</sup>θ*<sup>2</sup>

ð*<sup>θ</sup>*1*=*2*Tc*<sup>3</sup> �*θ*1*=*2*Tc*<sup>3</sup>

ð*<sup>θ</sup>*2*=*2*Tc*<sup>3</sup> �*θ*2*=*2*Tc*<sup>3</sup>

ð*<sup>θ</sup>*3*=*2*Tc*<sup>3</sup> �*θ*3*=*2*Tc*<sup>3</sup>

density j0 of the SC at T = 0 to meet this need.

*<sup>c</sup>*<sup>2</sup> *<sup>μ</sup>*<sup>1</sup> ð Þþ *<sup>λ</sup>*<sup>3</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p

ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p

ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p

∣*ξ*∣ þ ∣*W*3∣*=*2

∣*ξ*∣ þ ∣*W*3∣*=*2

<sup>3</sup>*=*<sup>2</sup> <sup>þ</sup>

*dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup>

*dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup>

*dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup>

values even when expressions under the radical signs are negative.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi **2***m=E<sup>F</sup>* <sup>p</sup> ð Þ *EF*>>*k<sup>θ</sup>*

ð*<sup>θ</sup>m=*2*Tc*<sup>3</sup> �*θm=*2*Tc*<sup>3</sup>

" #

" #

" #

ð*<sup>k</sup>θ<sup>m</sup>* �*kθ<sup>m</sup>*

1

1

1

*dx* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*kTc*3*x*þ*<sup>μ</sup>* <sup>p</sup>

θ<sup>m</sup> is the Debye temperature that has the highest value among all the Debye temperatures being used (when *λ*<sup>2</sup> 6¼ 0*, θ<sup>m</sup>* ¼ *θ*2*;* when *λ*<sup>2</sup> ¼ 0*, θ<sup>m</sup>* ¼ *θ*3*;* when *λ*<sup>2</sup> ¼ *λ*<sup>3</sup> ¼ 0*, θ<sup>m</sup>* ¼ *θ*1), and the operator *Re* ensures that the integrals yield real

If it turns out that we need to employ the *μ*-incorporated Eqs. (13)–(15) to deal with high-Tc SCs, rather than Eqs. (10)–(12), as will be shown to be the case, then there will be a need for at least one more equation for an additional property of the SC, besides equations for one of its Tc and Δs, in order to fix the values of μs and *λ*s that lead to various empirical features of the SC. We choose the critical current

The BSE-based approach enables one to derive an equation for j0 of an SC by employing the same values of *θ*s and *λ*s that determine a given value of its Tc and different values of its Δs. This comes about because one can now obtain expressions for the effective mass m\* of an electron, critical velocity v0 at T = 0, and the number of

" #

∣*ξ*∣ þ ∣*W*2∣*=*2

2

where μ<sup>0</sup> (μ1) is the chemical potential at T = 0 (T = Tc) and μ<sup>0</sup> has been used

ð*kθ<sup>m</sup>* �*kθ<sup>m</sup>*

*<sup>c</sup>*<sup>3</sup> *<sup>μ</sup>*<sup>1</sup> ð Þ¼ <sup>3</sup>

*, a*<sup>2</sup> *μ*<sup>0</sup> ð Þ¼ *Re*

*dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup>

*, b*<sup>2</sup> *μ*<sup>0</sup> ð Þ¼ *Re*

*dξ* ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup>

tanh ð Þ *x x*

tanh ð Þ *x x*

tanh ð Þ *x x*

4 *c*<sup>4</sup> *μ*<sup>1</sup> ð Þ � �1*=*<sup>3</sup>

> ð*kθ*<sup>2</sup> �*kθ*<sup>2</sup> *dξ*

<sup>p</sup> <sup>ð</sup><sup>1</sup> � *<sup>ξ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð*kθ*<sup>2</sup> �*kθ*<sup>2</sup> *dξ*

<sup>p</sup> <sup>1</sup> � *<sup>ξ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0

B@

*,*

<sup>1</sup>f g 1 � tanh ð Þ *x*

q

q

*,* (15)

ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p

∣*ξ*∣ þ ∣*W*2∣*=*2

3 7 5

ffiffiffiffiffiffiffiffiffiffiffiffiffi *ξ* þ *μ*<sup>0</sup> p

> 3 7 5

∣*ξ*∣ þ ∣*W*3∣*=*2

" #

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>W</sup>*<sup>2</sup> 2

" #

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>W</sup>*<sup>2</sup> 3

*,*

$$t(r, \boldsymbol{y}) = \left[ r \, \boldsymbol{y} \ln \left( \frac{r \, \boldsymbol{y}}{r \, \boldsymbol{y} - \mathbf{1}} \right) + \ln \left( r \, \boldsymbol{y} - \mathbf{1} \right) \right], \mathbf{r}\_i = \theta\_i / \theta.$$

Note that equating one (two) of the λs in Eq. (16) to zero, we obtain the equation for y in the 2PEM (1PEM) scenario.

In terms of y and the other parameters mentioned above, it has been shown in [19, 20] that

$$s \equiv m^\*/m\_e = A\_1 \frac{\left(\chi/\nu\_\text{g}\right)^{2/3}}{E\_F^{1/3}}\tag{17}$$

$$n\_s(E\_F) = A\_2(\chi/v\_\mathfrak{g})E\_F \tag{18}$$

$$P\_0(E\_F) = A\_3 \frac{\theta}{\mathcal{Y}} \frac{\left(\chi/\nu\_\text{g}\right)^{1/3}}{E\_F^{2/3}} \tag{19}$$

$$\upsilon\_0(E\_F) = A\_4 \frac{\theta}{\mathcal{Y}} \frac{1}{\left(\chi/\upsilon\_g\right)^{1/3} E\_F^{1/3}} \tag{20}$$

and hence

$$j\_0(E\_F) = \frac{n\_\text{s}(E\_F)}{2} e^\* \, v\_0(E\_F) = A\_5 \frac{\theta}{\mathcal{Y}} \left(\chi/v\_\text{g}\right)^{2/3} E\_F^{2/3}, \quad (e^\* = 2\varepsilon) \tag{21}$$

where *e* is the electronic charge, and

*<sup>A</sup>*<sup>1</sup> ffi <sup>3</sup>*:*305*x*10�<sup>10</sup> eV�1*=*<sup>3</sup> cm2K4*<sup>=</sup>*<sup>3</sup> *, A*<sup>2</sup> ffi <sup>2</sup>*:*729*x*10<sup>7</sup> eV�<sup>2</sup> K2 *, <sup>A</sup>*<sup>3</sup> ffi <sup>1</sup>*:*584*x*10�<sup>6</sup> cm K�1*=*<sup>3</sup> *, A*<sup>4</sup> ffi <sup>1</sup>*:*406*x*10<sup>8</sup> eV2*<sup>=</sup>*<sup>3</sup> sec �1K�5*=*<sup>3</sup> *,* and *<sup>A</sup>*<sup>5</sup> ffi <sup>6</sup>*:*146*x*10�<sup>4</sup> CeV�4*=*<sup>3</sup> K<sup>1</sup>*=*<sup>3</sup> s�<sup>1</sup>*:*

#### **2.5** *EF***-dependent equation for** *y* ¼ ð Þ *kθ=P***<sup>0</sup>** ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*m=E<sup>F</sup>* p

If the Tc and Δs of an SC are studied via Eqs. (10)–(12), which are valid when EF >> kθ, then it is appropriate to employ Eq. (16) for y because it too is valid when the same inequality is satisfied. If it turns out that we need to employ the μ-incorporated Eqs. (13)–(15) in lieu of Eqs. (10)–(12), then consistency demands that we employ for y, in lieu of Eq. (16), an equation that too is explicitly μ-dependent.

The desired equation for y in the 3PEM scenario, obtained by setting *P<sup>μ</sup>* ¼ ð Þ *E; P* in the BSE and doing away with the earlier constraint on EF, is [21]:

$$\mathbf{1} = \frac{\lambda\_1}{4} T(\Sigma\_1', r\_1 y) + \frac{\lambda\_2}{4} T(\Sigma\_2', r\_2 y) + \frac{\lambda\_3}{4} T(\Sigma\_3', r\_3 y), \tag{22}$$

constraint, was 20.4 meV corresponding to Tc = 95 K. Subsequently, even after employing μ-incorporated GBCSEs which are not constrained by inequality (8), we found that the 2PEM scenario was inadequate in explaining a value of Δ exceeding 20.4 meV [20, 21]. What remained elusive in these studies were especially the value Δ = 38 meV and the other values of Tc and Δ noted in Eq. (23). We show

*Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features…*

below how this lacuna is met via 3PEM in the present work.

*DOI: http://dx.doi.org/10.5772/intechopen.84340*

**4.2 GBCSEs constrained by inequality** *EF* ≫ *kθ***: premises and results**

Since working in the 2PEM scenario, both with the μ-independent and μ-dependent GBCSEs, the maximum value of *Δ* satisfying the Bogoliubov constraint was found to be 20.4 meV; we assume here that among the values noted in Eq. (23),

i.e., as our notation suggests, we attribute the first parameter above to 2PEM and

the other two parameters to 3PEM. With θs as given in Eq. (6), the solutions of Eqs. (10–12) lead to *λCa; λSr* f g *; λBi* ¼ f g 42*:*7*;* �37*:*6*;* 1*:*38 . Since these values are in conflict with the Bogoliubov constraint, in the absence of any other handle, we now systematically fine-tune the input values of the two Δs and Tc. We thus find that *λ*s

satisfying this constraint can be found only for values of Δs and Tc3 that are substantially different from their empirical values in Eq. (23). An example, in order

to obtain the set *λCa; λSr* f g¼ *; λBi* f g 0*:*4652*;* 0*:*2291*;* 0*:*4856 which satisfies the required constraint, we have to choose input values as Δ<sup>2</sup> ¼ 14*meV,* Δ<sup>3</sup> ¼ 33*:*5 *meV,* and *Tc*<sup>3</sup> ¼ 141 K*:*This is an unacceptable state of affairs, suggesting the need to have a handle other than variation of the Δs- and Tc-values of the SC to satisfy the Bogoliubov constraint. We show below how the μ-incorporated Eqs. (13)–(15), which are not constrained by inequality (8), meet this need.

**4.3 Interaction parameters obtained via μ-incorporated GBCSEs**

when

**65**

Before attempting to find the three λs satisfying the Bogoliubov constraint by employing Eqs. (13)–(15) for the same inputs as in Eqs. (6) and (25) and using μ as a handle, as a consistency check of these equations, we solve them for a value of *μ*<sup>0</sup> ≫ *kθ,* say, *μ*<sup>0</sup> ¼ 100 *k θ:* For this value of *μ*<sup>0</sup> ¼ *μ*1*,* or any other greater value, we find that *λCa* ¼ 42*:*69*, λSr* ¼ �37*:*59*,* and *λBi* ¼ 1*:*38*,* which are precisely the values we had obtained above via Eqs. (10)–(12) for the same inputs. This establishes that Eqs. (13)–(15) provide a reasonable generalization of Eqs. (10)–(12) and may therefore be used for any values of μ<sup>0</sup> and μ1. If we now solve Eqs. (13)–(15) with the same inputs as before, but with progressively lower values of μ<sup>0</sup> = μ1, we find that there is a marginal *increase* in the value of each of the λs; an example, for*μ*<sup>0</sup> ¼ *μ*<sup>1</sup> ¼ 5 *k θ, λCa* ¼ 42*:*90*, λSr* ¼ �37*:*78*, λBi* ¼ 1*:*39*:* This suggests that the assumption that μ<sup>0</sup> = μ<sup>1</sup> may not be valid. To check this, we solve our equations again with *μ*<sup>0</sup> ¼ 5 *k θ* and *μ*<sup>1</sup> ¼ 0*:*2 *μ*0*,*to find that *λCa* ¼ 16*:*46*, λSr* ¼ �15*:*56*,* and *λBi* ¼ 0*:*1144*:* It is thus confirmed that in adopting a low value of μ0, with μ<sup>1</sup> a small fraction of it, we are proceeding in the right direction. Following this course, we obtain the desired set of λ values corresponding to the Tc and Δ values in Eq. (25) as

*λCa; λSr* f g *; λBi* ¼ f g 0*:*3123*;* 0*:*4993*;* 0*:*5000 *,* (26)

*μ*<sup>0</sup> ¼ 1*:*95 k*θ* ð Þ 39*:*82 meV *, μ*<sup>1</sup> ¼ 0*:*105 *μ*<sup>0</sup> ð Þ 4*:*18 meV *:* (27)

Δ<sup>2</sup> ¼ 18 meV*,* Δ<sup>3</sup> ¼ 38 meV*,* Tc3 ¼ 95 K*,* (25)

where

$$T(\Sigma', y) = \text{Re}\left[\int\_0^1 dx \mathbf{F}(\Sigma', x, y)\right], \\ F(\Sigma', x, y) = f\_1(\Sigma', x, y) + f\_2(\Sigma', x, y)$$

$$f\_1(\Sigma', x, y) = 4\left[-u\_1(\Sigma', x, y) - u\_2(\Sigma', x, y) + u\_3(\Sigma', x, y) + u\_4(\Sigma', x, y)\right],$$

$$f\_2(\Sigma', x, y) = 2\ln\left[\frac{1 + u\_1(\Sigma', x, y)}{1 - u\_1(\Sigma', x, y)}\frac{1 + u\_2(\Sigma', x, y)}{1 - u\_1(\Sigma', x, y)}\frac{1 - u\_3(\Sigma', x, y)}{1 + u\_3(\Sigma', x, y)}\frac{1 - u\_4(\Sigma', x, y)}{1 + u\_4(\Sigma', x, y)}\right],$$

$$u\_1(\Sigma', x, y) = \sqrt{1 - \Sigma' x/y}, u\_2(\Sigma', x, y) = \sqrt{1 + \Sigma' x/y}$$

$$u\_3(\Sigma', x, y) = \sqrt{1 - \Sigma'(1 - x/y)}, u\_4(\Sigma', x, y) = \sqrt{1 + \Sigma'(1 - x/y)}$$

$$\Sigma' = k\theta/\text{E}\_2, \Sigma'\_i = k\theta\_i/\text{E}\_2, \ r\_i = \theta\_i/\theta.$$

Equating one (two) of the λs in Eq. (22) to zero, we obtain the μ-dependent equation for y in the 2PEM (1PEM) scenario.
