**4.4 Calculation of y via the μ-dependent Equation (22)**

A consistency check of Eq. (22): While both the μ-independent Eq. (16) and the μ-dependent Eq. (22) for y require as input the values of θs and λs, the latter equation requires the additional input of the value of EF [i.e., μ0, as noted below Eq. (15)] for its solution. We recall that Eq. (16) was derived by assuming that μ<sup>0</sup> >> k θ. Therefore if Eq. (16) is solved for any given values of θs and the λs, and Eq. (22) is solved with the same inputs together with a value of μ<sup>0</sup> which is much greater than k θ, then the solutions of both the equations should yield the same value for y. To carry out this test in detail, we solve Eq. (22) for values of θs as in Eq. (6) and those of λs as in Eq. (26) for the following seven cases: λ<sup>2</sup> = λ<sup>3</sup> = 0, λ<sup>3</sup> = λ<sup>1</sup> = 0, λ<sup>1</sup> = λ<sup>2</sup> = 0, λ<sup>1</sup> = 0, λ<sup>2</sup> = 0, λ<sup>3</sup> = 0, and *λ*<sup>1</sup> 6¼ *λ*<sup>2</sup> 6¼ *λ*<sup>3</sup> 6¼ 0*:* With the same values of the θs and λs and μ<sup>0</sup> = 100 k θ, solutions of Eq. (22) yield exactly the same results as were obtained via Eq. (16). Some of the values so-obtained are as follows: when λ<sup>2</sup> = λ<sup>3</sup> = 0 (1PEM scenario), y = 9.458; when λ<sup>2</sup> = 0 (2PEM scenario), y = 1.661; and when none of the *λ*s = 0 (3PEM scenario), y = 1. 195. Having thus shown that Eq. (22) is a reasonable generalization of Eq. (16) for arbitrary values of μ0, we now solve it for μ<sup>0</sup> (= EF) as given in Eq. (27) and values of {λ, θ} for each of the seven cases just noted. The results of these calculations are given in **Table 1**.

**4.5 The multiple values of the set** j*W*j ¼ Δ*; Tc; j*

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

validation.

**67**

Given in **Table 1** are also the values of the set j*W*j ¼ Δ*; Tc; j*

value of y noted in it, and the common values of γ, vg, and EF.

uents of j0 corresponding to each of the seven choices of λs mentioned above and the values of μs given in Eq. (27). Among these, the values of |W| and Tc are obtained via Eqs. (14) and (15), respectively. With EF (= μ0) fixed as in Eq. (27) and the values of γ and vg as noted in the legend for **Table 1**, s and ns are calculated via Eqs. (17) and (18), respectively. The values of v0 and j0 in each row of **Table 1** are obtained via Eqs. (20) and (21), respectively, with the input of θ = 237 K, the

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

Excepting |W| = 18 meV in the sixth row, and |W| = 38 meV and Tc = 95 K in the

last row of **Table 1**, which were employed as inputs to obtain the three λs, the values of all the other parameters in the Table are predictions following from our approach. Among these, Tc = 62.4 K, corresponding to |W| = 18 meV in the sixth row, is in excellent agreement with the experimental value, whereas the values of the subset {Tc (K), |W| (meV)} = {76.8, 26.3} in the fourth row are in reasonable agreement with the experimental values of {86, 28} [see Eq. (23)]. As concerns j0, we recall that the experimental value of 3.2 x 10<sup>7</sup> A cm�<sup>2</sup> noted for it in Eq. (24) corresponds to T = 4.2 K and H = 12 Tesla. Since our value of j0 = 7.3 x 10<sup>7</sup> A cm�<sup>2</sup> (in the last row of the Table) corresponds to T = H = 0, it too may be regarded as in reasonably good agreement with the experimental value. Since our

approach also provides values of the constituent parameters of j0, such as ns and v0, there is a need to monitor these parameters via experiment for its further

breakdown of one or the other mechanism responsible for pairing.

A brief account of the results following from Eq. (28) is as follows:

out above by taking the Δ- and the Tc-values of the SC as

**4.6 Are the results given in Table 1 stable?**

We have so far been concerned with the values of {Tc. Δ} as noted in Eq. (23). The significance of the additional values of Tc that GBCSEs have led to, as seen from **Table 1**, is as follows: these are the temperatures at which CPs bound via one- or more-phonon exchange mechanism breakup. Any such Tc corresponds to vanishing of the associated gap noted in the same row. One would thus expect to see, via appropriately sensitized experimental setups, some of these Tcs in the resistance vs. temperature plot of the SC. We believe that this expectation is met via the experimental results reported in [24] (reproduced in [25]) where, in a plot of resistance vs. temperature for the Bi-Sr-Ca-Cu-O system obtained via a setup sensitized to determine Tcs ≥ 75 K, there occur discontinuities at 75, 80, 85, 108, and 114 K. While these discontinuities are generally attributed to the presence of more than one phase in the SC, they may well alternatively/additionally be due to the

To address the above question, which is as relevant in the present context as it is for the numerical solution of any problem, we repeated the entire exercise carried

which deviate slightly from those considered earlier and noted in Eq. (25).

i. In order to satisfy the Bogoliubov constraint, we found that we needed *μ*<sup>0</sup> ¼ 2*:*4 *k θ* ¼ 49*:*02 meV ð Þ *θ* ¼ 237*K* and *μ*<sup>1</sup> ¼ 0*:*1 *μ*0*,*

Δ<sup>2</sup> ¼ 17*:*5 meV*,* Δ<sup>3</sup> ¼ 37*:*5 meV*,* Tc3 ¼ 94*:*5 K*,* (28)

**0**

**obtained via GBCSEs**

0 and the constit-


*|W| and Tc are obtained by solving Eqs. (14) and (15), respectively, with the input of μ<sup>0</sup> = 1.95 k θ (θ = 237 K), μ<sup>1</sup> = 0.105 μ0, different combinations of λ-values as given in Eq. (26), and the corresponding θ-values given in Eq. (6). For each set of {μ0, θ, λ}, y is obtained by solving Eq. (22). With* EF = μ0*, γ = (8/15) x10*�*<sup>3</sup> J, vg = 9.05 cm3 /g-at, and* s = m\*/me *= 4.97, vide Eq. (17), and ns = 4.0* � *<sup>10</sup>20/cm3 , vide Eq. (18). v0 and j0 corresponding to each value of y are obtained via Eqs. (20) and (21), respectively, with θ = 237 K.*

#### **Table 1.**

*Values of various superconducting parameters of Bi2Sr2CaCu2O8 obtained via GBCSEs.*

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

#### **4.5 The multiple values of the set** j*W*j ¼ Δ*; Tc; j* **0 obtained via GBCSEs**

Given in **Table 1** are also the values of the set j*W*j ¼ Δ*; Tc; j* 0 and the constituents of j0 corresponding to each of the seven choices of λs mentioned above and the values of μs given in Eq. (27). Among these, the values of |W| and Tc are obtained via Eqs. (14) and (15), respectively. With EF (= μ0) fixed as in Eq. (27) and the values of γ and vg as noted in the legend for **Table 1**, s and ns are calculated via Eqs. (17) and (18), respectively. The values of v0 and j0 in each row of **Table 1** are obtained via Eqs. (20) and (21), respectively, with the input of θ = 237 K, the value of y noted in it, and the common values of γ, vg, and EF.

Excepting |W| = 18 meV in the sixth row, and |W| = 38 meV and Tc = 95 K in the last row of **Table 1**, which were employed as inputs to obtain the three λs, the values of all the other parameters in the Table are predictions following from our approach. Among these, Tc = 62.4 K, corresponding to |W| = 18 meV in the sixth row, is in excellent agreement with the experimental value, whereas the values of the subset {Tc (K), |W| (meV)} = {76.8, 26.3} in the fourth row are in reasonable agreement with the experimental values of {86, 28} [see Eq. (23)]. As concerns j0, we recall that the experimental value of 3.2 x 10<sup>7</sup> A cm�<sup>2</sup> noted for it in Eq. (24) corresponds to T = 4.2 K and H = 12 Tesla. Since our value of j0 = 7.3 x 10<sup>7</sup> A cm�<sup>2</sup> (in the last row of the Table) corresponds to T = H = 0, it too may be regarded as in reasonably good agreement with the experimental value. Since our approach also provides values of the constituent parameters of j0, such as ns and v0, there is a need to monitor these parameters via experiment for its further validation.

We have so far been concerned with the values of {Tc. Δ} as noted in Eq. (23). The significance of the additional values of Tc that GBCSEs have led to, as seen from **Table 1**, is as follows: these are the temperatures at which CPs bound via one- or more-phonon exchange mechanism breakup. Any such Tc corresponds to vanishing of the associated gap noted in the same row. One would thus expect to see, via appropriately sensitized experimental setups, some of these Tcs in the resistance vs. temperature plot of the SC. We believe that this expectation is met via the experimental results reported in [24] (reproduced in [25]) where, in a plot of resistance vs. temperature for the Bi-Sr-Ca-Cu-O system obtained via a setup sensitized to determine Tcs ≥ 75 K, there occur discontinuities at 75, 80, 85, 108, and 114 K. While these discontinuities are generally attributed to the presence of more than one phase in the SC, they may well alternatively/additionally be due to the breakdown of one or the other mechanism responsible for pairing.

#### **4.6 Are the results given in Table 1 stable?**

To address the above question, which is as relevant in the present context as it is for the numerical solution of any problem, we repeated the entire exercise carried out above by taking the Δ- and the Tc-values of the SC as

$$
\Delta\_2 = \text{17.5 meV, } \Delta\_3 = \text{37.5 meV, } \text{T}\_{c3} = \text{94.5 K,} \tag{28}
$$

which deviate slightly from those considered earlier and noted in Eq. (25). A brief account of the results following from Eq. (28) is as follows:

i. In order to satisfy the Bogoliubov constraint, we found that we needed

 $\mu\_0 = 2.4$   $k\,\theta = 49.02 \text{ meV} \, (\theta = 237 \text{K}) \text{ and } \mu\_1 = 0.1 \,\mu\_0.$ 

**4.4 Calculation of y via the μ-dependent Equation (22)**

*On the Properties of Novel Superconductors*

**Table 1**.

**S. no.** *λ***1***λ***2***λ***<sup>3</sup>** *θ***1***θ***2***θ***<sup>3</sup>**

0 0

0.4993 0

0 0.5000

0.4993 0.5000

0 0.5000

0.4993 0

0.4993 0.5000

1 0.3123

2 0

3 0

4 0

5 0.3123

6 0.3123

7 0.3123

**Table 1.**

**66**

**(K)**

237 - -




237 - 269

237 286 -

237 286 269

*and* s = m\*/me *= 4.97, vide Eq. (17), and ns = 4.0* � *<sup>10</sup>20/cm3*

*of y are obtained via Eqs. (20) and (21), respectively, with θ = 237 K.*

∣*W*∣ **(meV)**

A consistency check of Eq. (22): While both the μ-independent Eq. (16) and the

μ-dependent Eq. (22) for y require as input the values of θs and λs, the latter equation requires the additional input of the value of EF [i.e., μ0, as noted below Eq. (15)] for its solution. We recall that Eq. (16) was derived by assuming that μ<sup>0</sup> >> k θ. Therefore if Eq. (16) is solved for any given values of θs and the λs, and Eq. (22) is solved with the same inputs together with a value of μ<sup>0</sup> which is much greater than k θ, then the solutions of both the equations should yield the same value for y. To carry out this test in detail, we solve Eq. (22) for values of θs as in Eq. (6) and those of λs as in Eq. (26) for the following seven cases: λ<sup>2</sup> = λ<sup>3</sup> = 0, λ<sup>3</sup> = λ<sup>1</sup> = 0, λ<sup>1</sup> = λ<sup>2</sup> = 0, λ<sup>1</sup> = 0, λ<sup>2</sup> = 0, λ<sup>3</sup> = 0, and *λ*<sup>1</sup> 6¼ *λ*<sup>2</sup> 6¼ *λ*<sup>3</sup> 6¼ 0*:* With the same values of the θs and λs and μ<sup>0</sup> = 100 k θ, solutions of Eq. (22) yield exactly the same results as were obtained via Eq. (16). Some of the values so-obtained are as follows: when λ<sup>2</sup> = λ<sup>3</sup> = 0 (1PEM scenario), y = 9.458; when λ<sup>2</sup> = 0 (2PEM scenario), y = 1.661; and when none of the *λ*s = 0 (3PEM scenario), y = 1. 195. Having thus shown that Eq. (22) is a reasonable generalization of Eq. (16) for arbitrary

values of μ0, we now solve it for μ<sup>0</sup> (= EF) as given in Eq. (27) and values of {λ, θ} for each of the seven cases just noted. The results of these calculations are given in

> **y** *v***<sup>0</sup>** x**10<sup>5</sup> (cms**�**<sup>1</sup> )**

1.70 8.5 9.688 1.41 0.9 1PEM

7.40 44.4 2.726 4.99 3.2 1PEM

7.01 28.5 2.886 4.72 3.0 1PEM

26.3 76.8 1.326 10.3 6.6 2PEM

17.4 54.1 1.671 8.15 5.2 2PEM

18.0 62.4 1.613 8.44 5.4 2PEM

38 95 1.199 11.4 7.3 3PEM

*|W| and Tc are obtained by solving Eqs. (14) and (15), respectively, with the input of μ<sup>0</sup> = 1.95 k θ (θ = 237 K), μ<sup>1</sup> = 0.105 μ0, different combinations of λ-values as given in Eq. (26), and the corresponding θ-values given in Eq. (6). For each set of {μ0, θ, λ}, y is obtained by solving Eq. (22). With* EF = μ0*, γ = (8/15) x10*�*<sup>3</sup> J, vg = 9.05 cm3*

*Values of various superconducting parameters of Bi2Sr2CaCu2O8 obtained via GBCSEs.*

*j***0** x**107 (Acm**�**<sup>2</sup> )** **Remark**

(Ca)

(Sr)

(Bi)

(Sr + Bi)

(Ca + Bi)

(Ca + Sr)

(Ca + Sr. + Bi)

*, vide Eq. (18). v0 and j0 corresponding to each value*

*/g-at,*

*Tc* **(K)** i. which led to

$$\{\lambda\_{Ca}, \lambda\_{Sr}, \lambda\_{Bi}\} = \{0.3628, 0.4338, 0.4918\},$$

pairing in them. By and large, this seems to be the currently popular view about

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

mean-field approximation is 3.49 � <sup>10</sup>�<sup>4</sup> eV]. (v) Some other results

<sup>7</sup>*:*84*x*1021 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>23</sup>*:*81 meV*; <sup>s</sup>* <sup>¼</sup> <sup>60</sup>*:*<sup>4</sup> <sup>≥</sup>*ns* cm�<sup>3</sup> ð Þ <sup>≥</sup>1*:*07*x*1020 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*324 meV*; <sup>s</sup>* <sup>¼</sup> <sup>253</sup> <sup>2</sup>*:*59*x*1022 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>25</sup>*:*34 meV*; <sup>s</sup>* <sup>¼</sup> <sup>126</sup> <sup>≥</sup>*ns* cm�<sup>3</sup> ð Þ≥3*:*31*x*1020 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*324 meV*; <sup>s</sup>* <sup>¼</sup> <sup>539</sup> *,*

It follows from even the brief account given above that the framework of μincorporated GBCSEs provides a viable explanation of the empirical features of CeCoIn5 as an alternative to the "popular view" about them; for a more detailed

where the upper row corresponds to pairing via the Ce ions and the lower row to

The salient superconducting properties of Ba0.6K0.4Fe2As2, which are generic of the Fe-based SCs (also called iron-pnictide SCs), are as follows (for sources of these properties, see [31]): (i) Superconductivity in this SC is due to the s�-wave state, which signifies that a gap below the Fermi surface and the one corresponding to it above the Fermi surface, Δ<sup>h</sup> and Δe, respectively, has opposite signs. (ii) In general, Δ<sup>h</sup> 6¼ Δe. (iii) Tc = 37 K. (iv) The highest Tc reported for this class of SCs > 50 K. (v) Prominent Δ-values: 6, 12 meV. (vi) Near-zero values of Δ: Δ for the SC can fall to zero along lines of its Fermi surface. (vii) Some other reported values of Δ (meV): 2.5, 9.0; 3.3, 7.6; 3.6, 8.5, 9.2; 4, 7, 12, 9.5. (viii) j*0*: exceeds 0.1 x 10<sup>6</sup> Acm�<sup>2</sup> at T = 4.2 K and H = 0; 1.1 x 10<sup>7</sup> Acm�<sup>2</sup> at T = 2 K and H = 0. (ix) A characteristic ratio for the SC: *EF=kTc* ¼ 4*:*4*:* (x) One of the s = m\*/me values: 9.0 (based on ARPES and a four-band model). (xi) Coherence length ξ = 9–14 Å. (xii) The Tc of

By adopting the framework of GBCSEs as outlined above, it has been shown in [31] that one can quantitatively explain *all* the above features of Ba0.6K0.4Fe2As2 without invoking a new superconducting state for the SC. This is an important remark because, on the basis of the multiband approach, it has been suggested in a recent review article [32] that superconductivity in Fe-based SCs is manifestation of

the SC plotted against a tuning variable has a dome-like structure.

*ns* <sup>¼</sup> <sup>1</sup>*=*3*π*<sup>2</sup> ð Þ <sup>2</sup>*mμ*1*=*ℏ<sup>2</sup> � �<sup>3</sup>*=*<sup>2</sup> h i are as follows:

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

treatment of the SC, we refer the reader to [30].

pairing via the Co ions.

**5.3 Fe-based SCs**

**69**

One of the reasons for regarding HFSCs as outside the purview of BCS theory is the conflict between inequality (8), which is a tenet of the theory, and inequalities (29) which the HFSCs are found to satisfy. Since μ-incorporated GBCSEs are not constrained by inequality (8), we employed them for a detailed study of CeCoIn5, which is a prominent member of the HFSC family. Salient features of this study are as follows. (i) The pairing mechanism was assumed to be 1PEM because we are dealing with a low value of Tc. (ii) Since 1PEM can be due to either Co or Ce ions, we considered both of these possibilities. (iii) Using Eqs. (4) and (5) and θ (CeCoIn5) = 161 K, we found θCe = 73 (276) K, corresponding to Ce as the lower (upper) bob in the double pendulum in the layer containing CeIn3, and θCo = 98.7, (294) K (in the layer containing CoIn3). (iv) Upon solving the μ-incorporated GBCSE for Δ, and different values of μ in an appropriate range determined via a consideration of inequalities (29), we were led to a multitude of Δ values in the range (2.76–3.49) � <sup>10</sup>�<sup>4</sup> eV corresponding to the single value of Tc = 2.3 K. For the SC under consideration, this is in qualitative accord with the experimental finding via the Bogoliubov quasiparticle interference (QPI) technique [the highest Δ value thus found is (5.5 � 0.05) � <sup>10</sup>�<sup>4</sup> eV, whereas our corresponding value based on the

these SCs.

i. and to values of y and j0 in the 1-, 2-, and 3-PEM scenarios as

$$\begin{aligned} \lambda\_2 = \lambda\_3 = \mathbf{0} \text{ (1PEM)}, \mathbf{y} = \mathbf{6.353}, j\_0 = \mathbf{1.58x10}^\top \text{ A/cm}^2\\ \lambda\_1 = \mathbf{0} \text{ (2PEM)}, \mathbf{y} = \mathbf{1.39}, j\_0 = \mathbf{7.21x10}^\top \text{ A/cm}^2\\ \lambda\_1 \neq \lambda\_2 \neq \lambda\_3 \neq \mathbf{0} \text{ (3PEM)}, \mathbf{y} = \mathbf{1.214}, j\_0 = \mathbf{8.24x10}^\top \text{ A/cm}^2.\end{aligned}$$

Since the sets of values given in Eqs. (25) and (28) lead to values of Δs, Tc, and j0 in the same ball park, it is seen that our results are stable with respect to small variations in the input variables.

## **5. Some other applications of GBCSEs**

### **5.1 La2CuO4**

La2CuO4 is unique not only because it heralded the era of high-Tc superconductivity but also because explanation of its Tc ≃ 38 K via GBCSEs requires invoking 2PEM, whereas it has only kind of ions (La) that can cause pairing. This paradoxical situation is resolved by appealing to the structure of the unit cell of the SC and employing Eqs. (4) and (5) to determine θLa. Since the unit cell comprises layers of LaO and OLa, if La is considered as the upper bob of the double pendulum in one of these, then it must be the lower bob in the other. Eqs. (4) and (5) therefore lead to two values of θLa. We thus have 2PEM in operation due to {λ, θLa1} and {λ, θLa2}. Following this approach [10], we were able to account for all the values of (2Δ0/KBTC), i.e., 4.3, 7.1, and 9.3, which were reported by Bednorz and Müller in their Nobel Lecture in 1987.

#### **5.2 Heavy fermion SCs (HFSCs)**

Superconductivity in HFSCs—so-named because a conduction electron in them behaves as if it has an effective mass up to three orders of magnitude greater than its free mass—was discovered by Steglich et al. [26]. Even though the highest Tc reported so far for this family of compounds is only 2.3 K (for CeCoIn5), these SCs have been the subject of avid study by theoreticians because of their following unusual properties:

$$E\_F < k\theta, kT\_c < E\_F, T\_c/T\_F \approx T\_F/\theta \approx \mathbf{0}.05,\tag{29}$$

where TF is the Fermi temperature. Besides the properties noted above, HFSCs are characterized by (a) large heat capacities of conduction electrons, much larger than those found for elemental SCs and about as large as those associated with fixed magnetic momenta, and (b) anisotropy of their gap structures. These features caused the term *exotic* or *unconventional* to be coined for them and led to the revival of an old idea that superconductivity may also arise due to the exchange of magnons, rather than just phonons. As a follow-up of this idea, it was shown in three well-known papers [27–29] that several experimental features of HFSCs can be explained if one assumes that magnetic fluctuations are the cause of d-wave

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

pairing in them. By and large, this seems to be the currently popular view about these SCs.

One of the reasons for regarding HFSCs as outside the purview of BCS theory is the conflict between inequality (8), which is a tenet of the theory, and inequalities (29) which the HFSCs are found to satisfy. Since μ-incorporated GBCSEs are not constrained by inequality (8), we employed them for a detailed study of CeCoIn5, which is a prominent member of the HFSC family. Salient features of this study are as follows. (i) The pairing mechanism was assumed to be 1PEM because we are dealing with a low value of Tc. (ii) Since 1PEM can be due to either Co or Ce ions, we considered both of these possibilities. (iii) Using Eqs. (4) and (5) and θ (CeCoIn5) = 161 K, we found θCe = 73 (276) K, corresponding to Ce as the lower (upper) bob in the double pendulum in the layer containing CeIn3, and θCo = 98.7, (294) K (in the layer containing CoIn3). (iv) Upon solving the μ-incorporated GBCSE for Δ, and different values of μ in an appropriate range determined via a consideration of inequalities (29), we were led to a multitude of Δ values in the range (2.76–3.49) � <sup>10</sup>�<sup>4</sup> eV corresponding to the single value of Tc = 2.3 K. For the SC under consideration, this is in qualitative accord with the experimental finding via the Bogoliubov quasiparticle interference (QPI) technique [the highest Δ value thus found is (5.5 � 0.05) � <sup>10</sup>�<sup>4</sup> eV, whereas our corresponding value based on the mean-field approximation is 3.49 � <sup>10</sup>�<sup>4</sup> eV]. (v) Some other results *ns* <sup>¼</sup> <sup>1</sup>*=*3*π*<sup>2</sup> ð Þ <sup>2</sup>*mμ*1*=*ℏ<sup>2</sup> � �<sup>3</sup>*=*<sup>2</sup> h i are as follows:

<sup>7</sup>*:*84*x*1021 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>23</sup>*:*81 meV*; <sup>s</sup>* <sup>¼</sup> <sup>60</sup>*:*<sup>4</sup> <sup>≥</sup>*ns* cm�<sup>3</sup> ð Þ <sup>≥</sup>1*:*07*x*1020 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*324 meV*; <sup>s</sup>* <sup>¼</sup> <sup>253</sup> <sup>2</sup>*:*59*x*1022 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>25</sup>*:*34 meV*; <sup>s</sup>* <sup>¼</sup> <sup>126</sup> <sup>≥</sup>*ns* cm�<sup>3</sup> ð Þ≥3*:*31*x*1020 ð Þ *<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*324 meV*; <sup>s</sup>* <sup>¼</sup> <sup>539</sup> *,*

where the upper row corresponds to pairing via the Ce ions and the lower row to pairing via the Co ions.

It follows from even the brief account given above that the framework of μincorporated GBCSEs provides a viable explanation of the empirical features of CeCoIn5 as an alternative to the "popular view" about them; for a more detailed treatment of the SC, we refer the reader to [30].

#### **5.3 Fe-based SCs**

i. which led to

*On the Properties of Novel Superconductors*

variations in the input variables.

their Nobel Lecture in 1987.

unusual properties:

**68**

**5.2 Heavy fermion SCs (HFSCs)**

**5.1 La2CuO4**

**5. Some other applications of GBCSEs**

*λCa; λSr* f g *; λBi* ¼ f g 0*:*3628*;* 0*:*4338*;* 0*:*4918 *,*

*<sup>λ</sup>*<sup>1</sup> 6¼ *<sup>λ</sup>*<sup>2</sup> 6¼ *<sup>λ</sup>*<sup>3</sup> 6¼ 0 3PEM ð Þ*, y* <sup>¼</sup> <sup>1</sup>*:*214*, j*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*:*24*x*10<sup>7</sup> <sup>A</sup>*=*cm2*:*

Since the sets of values given in Eqs. (25) and (28) lead to values of Δs, Tc, and j0

La2CuO4 is unique not only because it heralded the era of high-Tc superconductivity but also because explanation of its Tc ≃ 38 K via GBCSEs requires invoking 2PEM, whereas it has only kind of ions (La) that can cause pairing. This paradoxical situation is resolved by appealing to the structure of the unit cell of the SC and employing Eqs. (4) and (5) to determine θLa. Since the unit cell comprises layers of LaO and OLa, if La is considered as the upper bob of the double pendulum in one of these, then it must be the lower bob in the other. Eqs. (4) and (5) therefore lead to two values of θLa. We thus have 2PEM in operation due to {λ, θLa1} and {λ, θLa2}. Following this approach [10], we were able to account for all the values of (2Δ0/KBTC), i.e., 4.3, 7.1, and 9.3, which were reported by Bednorz and Müller in

Superconductivity in HFSCs—so-named because a conduction electron in them behaves as if it has an effective mass up to three orders of magnitude greater than its free mass—was discovered by Steglich et al. [26]. Even though the highest Tc reported so far for this family of compounds is only 2.3 K (for CeCoIn5), these SCs have been the subject of avid study by theoreticians because of their following

where TF is the Fermi temperature. Besides the properties noted above, HFSCs are characterized by (a) large heat capacities of conduction electrons, much larger than those found for elemental SCs and about as large as those associated with fixed magnetic momenta, and (b) anisotropy of their gap structures. These features caused the term *exotic* or *unconventional* to be coined for them and led to the revival

of an old idea that superconductivity may also arise due to the exchange of magnons, rather than just phonons. As a follow-up of this idea, it was shown in three well-known papers [27–29] that several experimental features of HFSCs can be explained if one assumes that magnetic fluctuations are the cause of d-wave

*EF* <*kθ, kTc* < *EF, Tc=TF* ≈*TF=θ* ≈0*:*05*,* (29)

i. and to values of y and j0 in the 1-, 2-, and 3-PEM scenarios as

*<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>3</sup> <sup>¼</sup> 0 1PEM ð Þ*, y* <sup>¼</sup> <sup>6</sup>*:*353*, j*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*58*x*107 <sup>A</sup>*=*cm2 *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> 0 2PEM ð Þ*, y* <sup>¼</sup> <sup>1</sup>*:*39*, j*<sup>0</sup> <sup>¼</sup> <sup>7</sup>*:*21*x*10<sup>7</sup> <sup>A</sup>*=*cm2

in the same ball park, it is seen that our results are stable with respect to small

The salient superconducting properties of Ba0.6K0.4Fe2As2, which are generic of the Fe-based SCs (also called iron-pnictide SCs), are as follows (for sources of these properties, see [31]): (i) Superconductivity in this SC is due to the s�-wave state, which signifies that a gap below the Fermi surface and the one corresponding to it above the Fermi surface, Δ<sup>h</sup> and Δe, respectively, has opposite signs. (ii) In general, Δ<sup>h</sup> 6¼ Δe. (iii) Tc = 37 K. (iv) The highest Tc reported for this class of SCs > 50 K. (v) Prominent Δ-values: 6, 12 meV. (vi) Near-zero values of Δ: Δ for the SC can fall to zero along lines of its Fermi surface. (vii) Some other reported values of Δ (meV): 2.5, 9.0; 3.3, 7.6; 3.6, 8.5, 9.2; 4, 7, 12, 9.5. (viii) j*0*: exceeds 0.1 x 10<sup>6</sup> Acm�<sup>2</sup> at T = 4.2 K and H = 0; 1.1 x 10<sup>7</sup> Acm�<sup>2</sup> at T = 2 K and H = 0. (ix) A characteristic ratio for the SC: *EF=kTc* ¼ 4*:*4*:* (x) One of the s = m\*/me values: 9.0 (based on ARPES and a four-band model). (xi) Coherence length ξ = 9–14 Å. (xii) The Tc of the SC plotted against a tuning variable has a dome-like structure.

By adopting the framework of GBCSEs as outlined above, it has been shown in [31] that one can quantitatively explain *all* the above features of Ba0.6K0.4Fe2As2 without invoking a new superconducting state for the SC. This is an important remark because, on the basis of the multiband approach, it has been suggested in a recent review article [32] that superconductivity in Fe-based SCs is manifestation of a *new* state. One of the reasons for this could well be the fact that the BCS equation for Δ is quadratic in Δ and is therefore unaffected when Δ ! �Δ. On the other hand, GBCSE for W1 is linear in this variable and has been derived by assuming that W1 undergoes a change in signature upon crossing the Fermi surface. This is also a feature of GBCSEs for W2, etc. Since s�-wave is an inbuilt feature of GBCSEs, one does not have to invent a new state for any SC, as has been suggested in [32].

#### **5.4 Isotope-like effect for composite SCs**

The following relation between Tc and the average mass of ions M in an elemental SC is well-known as the isotope effect:

$$T\_c \infty M^{-a}.\tag{30}$$

pairing. One then has the same λs in the equations for any Δ and the corresponding Tc of the SC—as is the case for elemental SCs. Multiple gaps arise in this approach because different combinations of λs operate on different parts of the Fermi surface due to its undulations. For a discussion of the moot question concerning the assumption of *locally* spherical values

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

On the other hand, the number of bands employed in MBA for the same SC differs from author to author. In this approach, in the two-band models, multiple gaps arise because the Hamiltonian is now postulated to have a term for pairing in each of the bands and another term corresponding to crossband pairing. For the Tc of the SC, one often employs the Migdal-Eliashberg-McMillan approach [36] which, even though based on 1PEM, allows λ to be greater than unity because it is based on an integral equation and the expansion parameter of which is not λ but me/M,

ii. Because of the feature of the GBCSE-based approach discussed above, with the input of the values of any two gaps of an SC and a value of Tc, it goes on to shed light on several other values of these parameters. This is not so for MBA.

i. Besides the SCs dealt with above, the GBCSE-based approach has been applied for the study of several elemental SCs, MgB2, YBCO and Tl-2212 [10, 19, 20], SrTiO3 [10], and NbN [21]. It is hence seen that a striking feature of this approach is its versatility: it is applicable to a wide variety of SCs that includes the so-called exotic or unconventional SCs such as the HFSCs and the Fe-based SCs, without invoking a new superconducting state for the latter.

ii. We believe to have shown that in explaining the empirical features of high-Tc SCs, the GBCSE-based approach goes farther than MBA or, so far as we are

(YBa2Cu3O7) > Tc (MgB2), etc., obviously suggests that the greater the complexity of structure of the SC, the greater is its Tc owing to the increase in the number of channels that may cause formation of CPs. While this is a situation that the GBCSE-based approach can easily deal with, in practice, going beyond Bi2Sr2Ca2Cu3O10, for example, is most likely to lead to an unstable and hence unrealizable SC. This is a problem that we believe belongs

iv. It was noted above that 14 different values of j0 have been reported for Bi-2212. For none of these values were reported the values of the other superconducting parameters of the SC. On the basis of [19, 20], we believe that, in order to help theory to suggest means to increase the Tc of an SC, the report of any of its properties, for example, j0, should be accompanied by, in so far as it is experimentally feasible, a list of all its other superconducting

v. The EF-incorporated GBCSEs in this communication correspond to the Tc of a composite SC and its Δ-values at T = H = 0. These equations can be further

iii. The fact that Tc (Bi2Sr2Ca2Cu3O10) > Tc (Bi2Sr2CaCu2O8) > Tc

characterizing the Fermi surface of a composite SC, see [33].

where M is the mass of an ion species.

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

aware, any other approach, e.g., [37].

to the realm of chemical engineering.

properties, viz., θ, Tc, Δs, m\*, v0, ns, γ, and vg.

**7. Conclusions**

**71**

While in the BCS theory α = 0.5, values significantly different from it have also been found for some elements such as Mo, Os, and Ru, for which α = 0.33, 0.2, and 0, respectively. Therefore, while Eq. (30) does not have the status of a *law*, it nonetheless helped in the formulation of BCS theory because it sheds light on the role of the ion lattice in the scenario of 1PEM, which has been shown to be the operative mechanism for elemental SCs.

We now draw attention to the fact that when Bi and Sr. in Bi2Sr2CaCu2O8 are replaced by Tl and Ba, respectively, the Tc of the SC increases from 95 to 110 K. Consequent upon these substitutions, the only property that can be unequivocally determined is the mass of the SC. Because the operative mechanism for pairing in composite SCs is not 1PEM, it becomes interesting to ask if Eq. (30) can be generalized to address the scenarios of 2PEM and 3PEM. A generalization of Eq. (30) that suggests itself naturally for the 2PEM scenario is

$$T\_c = p(\mathbf{M}\_1 \mathbf{M}\_2)^{-a},\tag{31}$$

where p is the constant of proportionality and M1 and M2 are the masses of ion species that cause pairing. Assuming that the value of Tc = 95 K for Bi2Sr2CaCu2O8 is due to 2PEM as in [25], an explanation for the increase in the Tc of Bi2Sr2CaCu2O8 when Tl2Ba2CaCu2O8 is obtained from it via substitutions was given in [33]. Noting that Bi and Tl belong to the same period and Sr. and Ba to the same group of the periodic table, several suggestions were made in [33] to further increase the Tc of the Bi-based SC. An example, following from our study, it was shown that Tc (Bi2 Mg2 CaCu2O8) should be 171 K.

#### **6. Discussion**

The Tcs and Δs of most of the hetero-structured, multi-gapped SCs which were studied above via the GBCSE-based approach have also been studied via the morewidely followed multiband approach (MBA) which originated with the work of Suhl et al. [34]. Because the former approach sheds light on additional features of these SCs, such as j0, s, ns, etc., as also because of its distinctly different conceptual basis, it complements the latter approach. Since the conceptual bases of both the approaches have been dealt with in detail in a recent paper [35], we confine ourselves here to the following remarks:

i. Employing the concept of a superpropagator, the GBCSE-based approach invariably invokes a λ for each of the ion species in an SC that may cause

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

pairing. One then has the same λs in the equations for any Δ and the corresponding Tc of the SC—as is the case for elemental SCs. Multiple gaps arise in this approach because different combinations of λs operate on different parts of the Fermi surface due to its undulations. For a discussion of the moot question concerning the assumption of *locally* spherical values characterizing the Fermi surface of a composite SC, see [33].

On the other hand, the number of bands employed in MBA for the same SC differs from author to author. In this approach, in the two-band models, multiple gaps arise because the Hamiltonian is now postulated to have a term for pairing in each of the bands and another term corresponding to crossband pairing. For the Tc of the SC, one often employs the Migdal-Eliashberg-McMillan approach [36] which, even though based on 1PEM, allows λ to be greater than unity because it is based on an integral equation and the expansion parameter of which is not λ but me/M, where M is the mass of an ion species.

ii. Because of the feature of the GBCSE-based approach discussed above, with the input of the values of any two gaps of an SC and a value of Tc, it goes on to shed light on several other values of these parameters. This is not so for MBA.
