**3. Superconducting features of Bi2Sr2CaCu2O8 addressed in this paper**

The reported empirical values of {Tc, Δ} of the above SC are [22], p 447:

$$\{T\_{\varepsilon}\left(\mathbf{K}\right), \Delta\left(\text{meV}\right)\} = \{\mathfrak{B}5, \mathfrak{A}\}, \{\mathfrak{A}6, \mathfrak{A}\}, \{\mathfrak{C}2, \mathfrak{A}\}.\tag{23}$$

As concerns empirical values of j0 of the SC, the situation is rather complicated because 14 values of this parameter have been reported in [22], p 468—depending upon the shape of the SC, how it is doped, and the values of temperature and the external magnetic field H under which they were determined. However, none of these values corresponds to the T = H = 0 scenario employed by us. The value *j* <sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>6</sup> <sup>A</sup>*=*cm2 at T = 4.4 K and unspecified value of H being closest to it. Another value of interest for us is [23]:

$$j\_0 = \text{3.2x10}^{\text{7}} \text{ A/cm}^2 \ (T = 4.2 \text{ K}, H = 12 \text{ Tesla}). \tag{24}$$

In the following we give a quantitative explanation of the values of the parameters noted in Eqs. (23) and (24) and a plausible explanation of the multitude of the other values of these parameters.

### **4. An explanation of the multiple {Tc, Δ, j0}-values of Bi-2212 via GBCSEs**

#### **4.1 Earlier work**

In our earlier work dealing with the above SC [10], we had employed GBCSEs constrained by inequality (8) and had restricted ourselves to the 2PEM scenario. The maximum value of Δ that we could then account for, subject to the Bogoliubov *Generalized BCS Equations: A Review and a Detailed Study of the Superconducting Features… DOI: http://dx.doi.org/10.5772/intechopen.84340*

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

### **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),

$$
\Delta\_2 = 18 \text{ meV}, \Delta\_3 = 38 \text{ meV}, \text{T}\_{c3} = 95 \text{ K}, \tag{25}
$$

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**

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

$$\{\lambda\_{Ca}, \lambda\_{Sr}, \lambda\_{Bi}\} = \{\mathbf{0}.\mathbf{3123}, \mathbf{0}.\mathbf{4993}, \mathbf{0}.\mathbf{5000}\},\tag{26}$$

when

$$
\mu\_0 = 1.95 \text{ k}\theta \ (39.82 \text{ meV}), \\
\mu\_1 = 0.105 \,\mu\_0 \ (4.18 \text{ meV}). \tag{27}
$$

The desired equation for y in the 3PEM scenario, obtained by setting *P<sup>μ</sup>* ¼ ð Þ *E; P*

*f* <sup>1</sup> Σ<sup>0</sup> ð Þ¼ *; x; y* 4 �*u*<sup>1</sup> Σ<sup>0</sup> ð Þ� *; x; y u*<sup>2</sup> Σ<sup>0</sup> ð Þþ *; x; y u*<sup>3</sup> Σ<sup>0</sup> ð Þþ *; x; y u*<sup>4</sup> Σ<sup>0</sup> ½ � ð Þ *; x; y ,*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � Σ<sup>0</sup>

ð Þ 1 � *x=y*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*x=y*

Equating one (two) of the λs in Eq. (22) to zero, we obtain the μ-dependent

**3. Superconducting features of Bi2Sr2CaCu2O8 addressed in this paper**

As concerns empirical values of j0 of the SC, the situation is rather complicated because 14 values of this parameter have been reported in [22], p 468—depending upon the shape of the SC, how it is doped, and the values of temperature and the external magnetic field H under which they were determined. However, none of these values corresponds to the T = H = 0 scenario employed by us. The value

<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>6</sup> <sup>A</sup>*=*cm2 at T = 4.4 K and unspecified value of H being closest to it. Another

In the following we give a quantitative explanation of the values of the parameters noted in Eqs. (23) and (24) and a plausible explanation of the multitude of the

In our earlier work dealing with the above SC [10], we had employed GBCSEs constrained by inequality (8) and had restricted ourselves to the 2PEM scenario. The maximum value of Δ that we could then account for, subject to the Bogoliubov

**4. An explanation of the multiple {Tc, Δ, j0}-values of Bi-2212 via**

The reported empirical values of {Tc, Δ} of the above SC are [22], p 447:

1 þ *u*<sup>2</sup> Σ<sup>0</sup> ð Þ *; x; y* 1 � *u*<sup>1</sup> Σ<sup>0</sup> ð Þ *; x; y* *λ*3 4 *T* Σ<sup>0</sup>

*, F* Σ<sup>0</sup> ð Þ¼ *; x; y f* <sup>1</sup> Σ<sup>0</sup> ð Þþ *; x; y f* <sup>2</sup> Σ<sup>0</sup> ð Þ *; x; y*

1 � *u*<sup>3</sup> Σ<sup>0</sup> ð Þ *; x; y* 1 þ *u*<sup>3</sup> Σ<sup>0</sup> ð Þ *; x; y*

q

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Σ<sup>0</sup>

1 þ Σ<sup>0</sup>

*x=y*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ 1 � *x=y*

� �

*, u*<sup>2</sup> Σ<sup>0</sup> ð Þ¼ *; x; y*

*, u*<sup>4</sup> Σ<sup>0</sup> ð Þ¼ *; x; y*

*<sup>i</sup>* ¼ *kθi=EF, ri* ¼ *θi=θ:*

f*Tc* ð Þ K *;* Δ ð Þ meV g ¼ f g 95*;* 38 *,* f g 86*;* 28 *,* f g 62*;* 18 *:* (23)

<sup>0</sup> <sup>¼</sup> <sup>3</sup>*:*2x107 <sup>A</sup>*=*cm<sup>2</sup> ð Þ *<sup>T</sup>* <sup>¼</sup> <sup>4</sup>*:*2 K*; <sup>H</sup>* <sup>¼</sup> 12 Tesla *:* (24)

<sup>3</sup>*;r*3*<sup>y</sup>* � �*,* (22)

1 � *u*<sup>4</sup> Σ<sup>0</sup> ð Þ *; x; y* 1 þ *u*<sup>4</sup> Σ<sup>0</sup> ð Þ *; x; y*

in the BSE and doing away with the earlier constraint on EF, is [21]:

*dxF* Σ<sup>0</sup> ð Þ *; x; y* � �

1 � *u*<sup>1</sup> Σ<sup>0</sup> ð Þ *; x; y*

1 � Σ<sup>0</sup>

q

Σ<sup>0</sup> ¼ *kθ=EF,* Σ<sup>0</sup>

*λ*2 4 *T ;* Σ<sup>0</sup> <sup>2</sup>*;r*2*<sup>y</sup>* � � <sup>þ</sup>

<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>1</sup> 4 *T* Σ<sup>0</sup> <sup>1</sup>*;r*1*<sup>y</sup>* � � <sup>þ</sup>

*On the Properties of Novel Superconductors*

*<sup>f</sup>* <sup>2</sup> <sup>Σ</sup><sup>0</sup> ð Þ¼ *; <sup>x</sup>; <sup>y</sup>* 2 ln <sup>1</sup> <sup>þ</sup> *<sup>u</sup>*<sup>1</sup> <sup>Σ</sup><sup>0</sup> ð Þ *; <sup>x</sup>; <sup>y</sup>*

*u*<sup>3</sup> Σ<sup>0</sup> ð Þ¼ *; x; y*

value of interest for us is [23]:

*j*

other values of these parameters.

**GBCSEs**

**4.1 Earlier work**

**64**

*u*<sup>1</sup> Σ<sup>0</sup> ð Þ¼ *; x; y*

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

q

ð1 0

*T* Σ<sup>0</sup> ð Þ¼ *; y Re*

where

*j*
