Heavy-fermion superconductors (HFSCs) [1] belong to a fascinating family of compounds which can also be found in a variety of other states such as metallic, insulating and magnetic. HFSCs are so called 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. Superconductivity in these compounds was discovered by Steglich et al. [2] ; since then it has been authoritatively dealt with, e.g., in [3] and [4] .

Some remarkable features of HFSCs are [5] :

where E_F, θ_D, T_c, and T_F denote, respectively, the Fermi energy, Debye temperature, critical temperature and Fermi temperature of the SC, and k_B is the Boltzmann constant.

Note in particular that Inequality (1a) is in conflict with the following basic assumption of BCS theory

where μ is the chemical potential. Therefore, despite their rather low T_cs, HFSCs are regarded as outside the purview of the usual BCS equations. This feature―at least partly―is responsible for coinage of the term exotic or unconventional for them.

HFSCs are characterized by two additional distinguishing features that have a bearing on the pairing mechanism operative in them. These are: 1) large heat capacities of conduction electrons―much larger than those found for the elemental SCs and as large as those usually associated with fixed magnetic momenta, and 2) anisotropy of their gap-structures. These features naturally led to the revival of an old idea that superconductivity may also arise without the mediating role of phonons. Thus, with He-3 in mind, triplet pairing was initially regarded as the cause of superconductivity in these SCs. However, it was subsequently shown in three well-known papers [6] -[8] that several experimental features of HFSCs can be explained if one assumes that magnetic fluctuations are the cause of d-wave pairing in them. By and large, this currently seems to be the popular view.

The study undertaken here is not in a spirit of dissension with the above view. Rather, it is motivated by the question: does the “popular” view definitively exclude all other mechanisms of pairing? We believe it does not because HFSCs are materials that exhibit multi-phase, multi-scale complexity. These features―together with the fact that at least one of such SCs, namely UPt₃, is already known to undergo more than one phase transition―suggest that an HFSC may well be characterized by different forms of pairing mechanisms in different parts of momentum space. This is a scenario inspired by the multi-condensate superconductivity approach advocated by Bianconi and collaborators ([9] , and references therein). We now recall that a case has been made for the applicability of the electron-lattice interactions to a wide variety of SCs [10] ; also that by employing such interactions, the T_cs and multiple gaps of several high-T_c SCs have been accounted for via the generalized BCS equations [11] -[14] . Note that complexity of structure is a feature common to both high-T_c SCs and HFSCs.

Guided by above considerations, we assume here that itinerancy of conduction electrons and hybridzation of orbitals in an HFSC can cause them to be in a region of momentum space where, effectively, pairing is via an s-wave electron-lattice interaction. In order then to deal quantitatively with an HFSC, we need equations for its T_c and ∆₀ (the gap at T = 0) that are not constrained by Inequality (2), i.e., equations that contain μ as a variable. Such equations, which are in fact an integral part of BCS theory, have been available since at least the time of [15] concerned with BCS-BEC crossover physics. A feature of these equations that seems to us to have escaped serious attention is: after suitable modification as discussed below, they can also be employed to shed light on real life SCs such as HFSCs.

This paper is organized as follows. In the next section we present the μ-incorporated equations for ∆₀ and T_c― equations which are variants of those considered in [15] and were recently obtained in studies concerned with 1) crossover physics without appeal to scattering length theory [16] and 2) a long-standing puzzle posed by superconducting SrTiO₃ [17] . Since these equations have a greater range of applicability than the usual equations sans μ, they may be regarded as comprising an extended form of BCS theory. In the following we shall therefore refer to them collectively as ET―extended (BCS) theory. ET is applied in Section 3 (4) to a prominent member of the HFSC family, namely CeCoIn₅, by assuming that pairing in it is caused by Ce(Co) ions. Section 5 deals with densities of the charge carriers that our study leads to. This is followed up by a detailed discussion of our approach and a succinct summing up.

BCS equation for ∆₀ and T_c incorporating μ were obtained in [17] as:

where

and V (as in) is a term due to both Coulomb repulsion between a pair of electrons and attraction between them because of the ion-lattice. It is easily seen that Equations (3) and (4) go over to the usual BCS equations for ∆₀ and T_c, respectively, when Inequality (2) is imposed.

Equations (3) and (4) have been written here―unlike in [17] ―by assuming that

where sub-scripts 0 and 1 denote values of a symbol at T = 0 and T = T_c, respectively. Strictly speaking, i.e., to be perfectly general, distinction should be made between V₀ and V₁, as also between values of each of the other two variables at the two temperatures. However, as was pointed out in [17] , on doing so one is led to an undetermined set of equations. We recall that neither V nor μ is required for the calculation of either ∆₀ or T_c via the usual BCS equations―for both of which λ has the same value. Therefore, as a matter of fact, the final subset of equalities in the set of Equations (11) is in accord with a tenet of the BCS theory.

An important feature of ET is that its expressions for, and given by Equations (5), (6) and (7), respectively, lead to complex values when μ < k_Bθ_D. Real solutions can however still be obtained by replacing the lower limit (−k_Bθ_D) in each of these equations by ?μ. It is in this sense that the word modification was used above. While the issue of such solutions will be further discussed below, we shall henceforth refer to any equation with curtailed limits as a truncated equation.

We now apply ET to CeCoIn₅ which has the highest T_c in its family. Its characteristic features of immediate concern to us are:

[18] ; [19] . (12)

It follows from the structure of CeCoIn₅ that, in the one-phonon exchange scenario, pairing in it can be caused by either of Ce and Co ions. We first consider Ce ions.

We have so far considered the effects of pair-formation via the Ce ions. On repeating the above exercise with Co ions, we find that θ_D(Co) = 98.7 K in In₂Co layers, i.e., when Co is lower of the two bobs in the double pendulum; otherwise θ_D(Co) = 294.1 K (in CoIn₂ layers). On now solving Equations (4) and (3), the plots of λ and ∆₀ against μ are found to be similar to those obtained earlier and hence are not included here. Given in Table 2 are all the other results akin to those that were given in Table 1 for pairing via the Ce ions. We note that the value of v(Co) is 6.67 cm³, which differs considerably from that of v(Ce) given in Equation (20).

Density of charge carriers, n, is an important parameter for any SC. It is given by

In this equation m is to be taken as s times the free electron mass. The results for n following from the s and values given in the two tables are as follows:

For pairing via the Ce ions:

For pairing via the Co ions:

Based on experiments performed by employing the Bogoliubov quasi-particle interference (QPI) technique, it has recently been reported [22] that CeCoIn₅ is characterized by a multitude of gap-values, the greatest among

these being (5.5 ± 0.05) × 10⁻⁴ eV. This value has been specifically attributed to the region of pairing.

The investigation reported here too has led to a multitude of ∆₀ values―in the (2.76 - 3.49) × 10⁻⁴ eV range for pairing via the Ce ions with θ_D = 73.3 K―corresponding to a single T_c value. The fact that our maximum value is lower than the maximum value reported in [22] is not surprising in view of the mean-field approximation on which it is based.

That there must be an attribute of CeCoIn₅ which causes it to have the above features (single T_c, multiple gaps) is obvious. By and large, μ is believed to be that attribute because ∆₀ is supposed to follow its fluctuations. The new feature of our approach is the suggestion that HFSCs be addressed by incorporating μ into the BCS equations for T_c and ∆₀. Implementation of this suggestion led us to consideration of four equations: the μ-incorpo- rated BCS equation for T_c and the corresponding number equation, besides the μ-incorporated BCS equation for ∆₀ and the number equation at T = 0. While the latter two equations have provided the starting point of all papers that have addressed the problem of the BCS-BEC crossover physics, what seems to us to have escaped serious attention is that, upon modification, they can also be used to account for the properties of HFSCs. While indeed modification requires truncation of the equations when μ < k_Bθ_D, we recall that such a need also arises for the same reason (i.e., to avoid solutions with complex values) in the study of 1) the crossover physics without appeal to scattering length theory [16] , 2) various Bose condensates via a linearly perturbed harmonic oscillator potential [23] , and 3) the puzzle posed by the dilute SC SrTiO₃ [17] . Note however that even though the use of truncated equations for CeCoIn₅ here and SrTiO₃ earlier is a common feature of our approach, the values of μ and the exact amounts by which the limits of these equations are curtailed are different in the two cases because of the differences in the structures of the SCs. Apart from the rather straightforward mathematical requirement of avoiding solutions with complex values, truncation of the BCS equations can be justified on physical grounds by appealing to itinerancy of electrons. This much-invoked concept was perhaps first used by Suhl et al. [24] in the context of multi-band superconductivity. Since then it has been used routinely for multi-gap superconductors including HFSCs. It was noted in [17] that itinerancy of electrons can cause them to be in a region akin to a “rain-shadow” region where the entire spectrum of phonon energies is not available to them for pairing. We give below an analogy by way of further elaboration.

Imagine a convoy on a road passing through a range of mountains. As the road twists and turns through a series of valleys and mountain-tops, the amount of sunlight the convoy receives will vary from a maximum at the highest point of the range to a minimum at a place determined by its topography. Since 3-d Fermi surfaces of HFSCs are known to have many sheets and highly complex structures, they may be thought of as providing a similar backdrop for the streams of conduction electrons. It then follows that there will be places on the Fermi surface where itinerant conduction electrons can exchange phonons with either the A- or the B-ions separately, besides a place or places where they can exchange phonons with both of them simultaneously; also that the phonon energies available to them for pairing in these places may well be restricted.

Implicit to our approach is the assumption of a locally spherical Fermi surface [LSFS] for each value of μ that we have used. We should like to stress that this is similar to what is done for elemental SCs which too have rather complex structures, as can be seen in [25] . The difference between an element and any composite material is that the latter is characterized by more than one mean value of μ. Hence the approach followed here may be viewed as modeling of a very complex situation in a rather simple (if not the simplest possible) manner. It seems interesting to observe that the concept of LSFSs is reminiscent of the concept of locally inertial coordinate frames employed in the general theory of relativity [26] .

For low values of μ, the difference between these and the corresponding values of and in the two tables is rather pronounced. This warrants a comment because not only are μ and E_F generally used interchangeably in the superconductivity literature, but also because no distinction is made between the values of E_F at T = 0 and T = T_c. While both E_F and μ signify the amount of energy required to add one particle to the existing many-particle system, strictly speaking, the former is a concept applicable to a non-interacting system whereas the latter applies to an interacting system. Since temperature lifts the degeneracy of a non-interacting system of fermions at T = 0, it follows that should be greater than, as is also seen from the two tables. Further, because interactions cause a reduction in the number of free electrons because of pairing, it follows that μ should be less than both and, as is also reflected in the two tables. While we have so far talked about low values of μ, we note that the inequalities hold for any value of μ, as can be seen even from the top row in Table 1. We also note that for, the differences between them show up in the fifth or sixth places after the decimal.

We now draw attention to the two figures in this paper. The first of these has a dome-like structure, which implies that one can have the same value of λ for two different values of μ. The import of the second figure is that the same value of λ―when combined with different values of μ―leads to different values of ∆₀. We note that the parabolic feature of Figure 2 brings out that the variation in ∆₀ for decreasing values of μ (for T = T_c) is similar to its variation for increasing values of T (when).

We should finally also like to draw attention here to a study of the de Haas-van Alphen oscillations in the BCS theory that took into account an external magnetic field via the Landau quantization scheme [27] , but not the chemical potential. A further extension of ET that incorporates an external magnetic field may well lead to additional insights about HFSCs.

We have shown here that even the so-called exotic or unconventional SCs can be brought into the purview of an extended form of BCS theory which is based significantly on ideas due to Born and Karmann [20] , Suhl et al. [25] , and Bianconi et al. [9] . The application of the modified theory to CeCoIn₅ has shed light not only on its T_c(s) and multiple gaps, but also on the mass values of its heavy electrons and their number densities.

The author acknowledges that participation in QCM14 at Obergurgl, Austria, stimulated his interest in the superconductivity of HFSCs. He is grateful to the Organizers of this Workshop for their support enabling his participation. He thanks Professor V. S. Varma for a valuable discussion and for verification of some of the numerical work. He also acknowledges that his correspondence with Dr. Eagles and Professor M. de Llano was valuable in this study. He would specially like to thank Mr. Ashwani Kumar, School of Physical Sciences, Jawaharlal Nehru University, for assistance in connection with some of the numerical work. Finally, he thanks Professor D. C. Mattis for encouragement.

G. P.Malik, (2015) A Study of Heavy-Fermion Superconductors via BCS Equations Incorporating Chemical Potential. Journal of Modern Physics,06,1233-1242. doi: 10.4236/jmp.2015.69128