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**Employing the**** ****Geilikman-Kresin (GK) theory, we address the experimental data obtained by Bauer et al., and by Schneider et al.,**** ****on the thermal conductivity (κ) of superconducting MgB _{2}. The two gaps of this compound have qualitatively been understood via the well-known Suhl, Matthias, and Walker’s (SMW) approach to multi**

**gap superconductivity. Since this approach is based on one-phonon exchange mechanism for the formation of Cooper pairs, it cannot give a quantitative account of the values of T**

_{c}and the multiple gaps that characterize MgB_{2}and other high-T_{c}superconductors (SCs). Despite this fact and some rather ambiguous features, it has been pointed out in a recent critical review by Malik and Llano (ML) that the SMW approach provides an im**portant clue to deal with an SC the two gaps of which close at the same T**

_{c}**:**

**consider the possibility of the interaction parameters in the theory to be temperature-dependent. Guided by this clue, ML gave a complete summary of parameters that quantitatively account for the T**

_{c}and the gaps of MgB_{2}via the generalized BCS equations (GBCSEs). GBCSEs which we recall, invoke multi-phonon exchange mechanism for the formation of Cooper pairs and multiple Debye temperatures to deal with composite SCs. The parameter-values given in ML are used here to calculate the temperature-dependent gaps, which are an essential input for the GK theory. Notable features of this work are:**1**

**) k**

_{MgB2}**is calculated for both—the scenario in which the two gaps of MgB**

_{2}close/do not close at the same temperature whence it is found**that**

**2**

**) the latter scenario yields result**

**s**

**in better agreement with experiment.**

Thermal conductivity (κ) of a superconductor (SC) is an important parameter from the point of view of applications; additionally, it helps in the theoretical understanding of the superconducting state [_{c} to its maximum value at T = 0—is synonymous with a similar increase in the number of density of Cooper pairs (CPs) and hence with a decrease in the number density of heat carriers because CPs have zero entropy.

The existence of a gap in an SC at any T thus profoundly affects (lowers) both, the number density of the heat carriers (the unpaired electrons) and their specific heat. If κ of an SC at any T does not have a value as low as one might expect on these grounds, it is so because the lower the T is, the greater the MFP is. There is another feature that affects κ: heat is transported not only by electrons, but also by phonons—the quanta of lattice vibrations. The determination of κ due to any of these carriers requires the solution of a Boltzmann equation taking into account the nature of the dominant scattering processes appropriate for a given sample. If one is dealing with a composite superconductor (CS), there are additional complicating factors. Calculation of κ(T) of a CS is thus a rather formidable problem; we refer the reader to the lucid review by Uher [

The basic equations that yield the thermal conductivity of an SC based on the BCS theory [_{es}, the electronic thermal conductivity of a simple superconductor when the scattering of electrons by impurities is the dominant process, the equation given by GK determines κ_{gs}, the thermal conductivity due to phonons when phononelectron scattering dominates. Not surprisingly, one finds that an application of these equations requires knowledge of ∆(T) of the SC for all T ≤ T_{c}. We note that equations similar to the G and the GK equations were also derived by Bardeen, Rickayzen and Tewordt (BRT) [

In this paper, we address the experimental data on the thermal conductivity of MgB_{2} obtained by Bauer (B) et al. [_{2}. Employing GBCSEs to calculate the values of these gaps for, we attempt here to shed light on their role, individually and collectively, in determining the total thermal conductivity of the compound. We note that in the earlier work [10-12] in which GBCSEs were used for a variety of high-temperature superconductors (HTSCs), the gap-values were calculated only at T = 0.

An experimental feature of MgB_{2} as reported by, e.g., Iavarone et al. [_{c} and the gap-values of the high-T_{c} SC (HTSC) MgB_{2}? Since the approach was originally given in the context of transition elements, it can only be adapted for composite SCs such as MgB_{2} which has no d-electrons. Such attempts have of course been made. Thus, making out a case for going beyond even the Eliashberg superconductivity, Liu et al. [_{2} to the multiple band structure of its Fermi surface that brings into play different phonon modes; in particular, based on density-functional calculations invoking twophonon exchange mechanism they arrived at the value of (clean limit) for the effective electron-phonon coupling constant which, they concluded, “is arguably consistent with the measured T_{c} of nearly 40 K.” In another appeal to the SMW approach in the context of MgB_{2}, Choi et al. [_{c} of MgB_{2} were attributed to the existence of two separate populations of electrons—nicknamed “red” and “blue,” leading them to note: “Stated differently, electrons on different parts of the Fermi surface form pairs with different binding energies.”

Notwithstanding the above developments, the SMW approach has till now not led to a framework in which the T-dependent gaps and T_{c} of an HTSC may be calculated. However, it seems interesting that it should have led to such concepts as two-phonon exchange mechanism [_{c}, the clue being: consider also the possibility of the interaction parameters in the theory to be T-dependent. These are precisely the concepts that are manifestly incorporated in the Bethe-Salpeter equation (BSE)-based approach that led to GBCSEs (Approach 2 hereafter); this approach has already been shown to deal quantitatively with the multiple gaps and T_{c}s of a variety of HTSCs with a measure of success [10-12]. It is worth noting that each interaction parameter in Approach 2 satisfies the Bogoliubov constraint for the BCS theory given by

as discussed in [

The paper is organized as follows. In the next section, we give an account of GBCSEs and the G and the GK equations which form our framework. Since it was reported in [_{2} close at about 40 K, we consider in Section 3 the experimental data of both the B [_{2} close at different temperatures (Scenario 2). Interestingly, it is found that whereas both the scenarios lead to almost indistinguishable results for, the latter scenario yields results in better agreement with experiment for t < 0.448. To avoid repetition, we present here the results of our calculations pertaining to the data of only B’s group. Our findings pertaining to the data of the other group are of course summarized. The final section sums up our findings.

Our framework has three constituents. The first of these is concerned with calculating the values of ∆_{1}(T) and ∆_{2}(T) of MgB_{2} for all via GBCSEs in both the scenarios mentioned above. We first deal with Scenario 2 in which the interaction parameters are T-independent and recall that the BCS equation for ∆(T) for a simple SC is:

The GBCSE equivalent to (2) is [9,19]:

or

where β = 1/k_{B}T, k_{B} being the Boltzmann constant, Θ the Debye temperature, and 2|W_{1}| is the binding energy of a CP that is bound via one-phonon exchanges (recall that the energy required to break a CP is 2∆). Note that putting W_{1} = 0 in (3) or ∆ = 0 in (2) yields the familiar BCS equation for T_{c} of the SC. This already suggests a connection between W and ∆. Further, it is readily seen that when T = 0 (tanh = 1) and λ→0, both (2) and (3) yield:.

The generalized version of (3) when CPs in a CS are bound via two-phonon exchanges is [

or

where is the interaction parameter due to the i^{th} species of ions in the CS, to be distinguished from λ_{i}, which denotes the interaction parameter of the same species in its free state; a similar distinction applies to and Θ_{i}; 2|W_{2}| is the binding energy of a CP bound via two phonon exchanges. The equation for T_{c} of the CS in this case follows by putting W_{2} = 0 in (4a):

The two Debye temperatures in the problem cater to the anisotropy of the CS [

The framework of GBCSEs [_{2} is [

These lead via (4a) and (5) to

The parameters corresponding to B alone in (6) lead via (3a) and the BCS equation for T_{c} (Equation (5) with) to

Hence we use |W_{1,2}| and |∆_{1,2}| interchangeably from now on; the T-dependent values of these can be calculated via (3a) and (4a). Note that both the λs in (6) satisfy the Bogoliubov constraint given in (1) above.

If experiment dictates that the smaller gap of MgB_{2} also closes at about 39 K (and not at about 13 K)—as has been reported in [_{B}T_{c} = 3.53. For a detailed discussion of how the SMW approach also implies such dependence of the interaction parameters, we refer the reader to [_{2} is met by replacing above as follows:

We now turn to the second constituent of our framework: the G equation for κ_{es}(T) when electrons are scattered predominantly by impurities. This equation is [

where is independent of T,

and

.

We note that [

The final constituent of our framework is the GK equation for κ_{gs} when the phonons are scattered predominantly by electrons. This equation is [

where is independent of temperature, and

As was the case for (11), [

We first address the data under consideration in Scenario 1, i.e., when both the gaps close at the same T_{c}. To this end we solve (3a) and (4a) for ∆_{1}(T) and ∆_{2}(T) with - values taken from (6) and expressions for λ(T) as given in (8); the results are given in _{es}(T, ∆) and F_{gs}(T, ∆) via (10) and (12), respectively. This exercise is carried out at each of the 38 temperatures below T_{c} for which the experimental values of the total thermal conductivity κ_{s} are given in the data of B et al.

[

Using (9) and (11), we now write

as

where denotes reduced thermal conductivity, t = T/T_{c}, and A, B, f_{es} and f_{gs} are dimensionless.

Equation (14) pertains to the situation when the SC is characterized by one gap; when there are two gaps, we have

We now need to reduce the four F’s in

To proceed further we need to fix A and B. We do so by appealing to the experimental values of κ_{s} at two temperatures. This enables us to calculate κ_{s} at the remaining 36 temperatures.

The above calculations are repeated for Scenario 2. Given below are the values for A and B found at different combinations of temperatures in the two scenarios (the values in the parentheses correspond to Scenario 2):

In each scenario, the results for κ_{s}(T) yielded by all of the above pairs of A, B values are similar. The values adopted by us are those given in the top set. While we have thus calculated κ_{s}(T)s at each of the remaining 36 temperatures in the data under consideration, we have given in _{s}(T)│_{theory} (and its constituents) in the two scenarios—together with the plot of κ_{s}(T)│_{exp}— plotted against the reduced temperature.

We have also analyzed the data of S et al. [_{1} = 0.452423 K and T_{2 }= 25.84772 K, we obtain: . _{s}(T)│_{exp} and κ_{s}(T)│_{theory} against some selected reduced temperatures for this case.

1) While it has been reported [_{2} close at the same T_{c} (Scenario 1), this result does not pertain to the conditions under which thermal conductivity is determined. For this reason we chose here to also address the experimental data in the additional scenario in which the gaps close at different T_{c}s (Scenario 2). In both cases, as can be seen from _{es}(T, ∆_{1}) and κ_{es}(T, ∆_{2}) decrease with T, near T_{c},. Further, at the lowest temperature in the table, R ≈ 60 (Scenario 1) and R ≈ 115 (Scenario 2). Thus: a) in the entire range of temperature to which the experimental data of B et al. [_{gs}(t) makes negligible contribution to κ_{s}(t) in both the scenarios, b) in the range the results in the two scenarios differ by no more than 3%, and c) for t < 0.4477, κ_{s}(t)-values in Scenario 1 are lower than in Scenario 2—by about 40% at t = 0.2684, for example. Overall, the latter scenario is thus found to be in better agreement with experiment 2) In order to shed light on the above findings, we draw attention to the following relations obtained via the Gorter-Casimir two-fluid theory of superconductivity, as in, e.g., [

where N_{s} is the density of superconducting electrons. So far as the temperature dependence in (16) is concerned, it has been noted that “some authors report other exponent values or related expressions” [

The second relation in (16) is strikingly similar to

We thus infer that greater the value of δ(t) greater is the fraction of superconducting electrons (or CPs) and therefore smaller the fraction of available electrons as heat-carriers. It is only under this circumstance that heat is predominantly carried by phonons, i.e., when t is close to 0 K. This explains the result in para 1 (a) in this section: the contribution of κ_{gs}(t) to κ_{s}(t) is negligible be-

cause bulk of the data under consideration pertains to values of t not close enough to 0 K. Note that in both the scenarios the Sp. Ht. and the MFP of electrons have the same values at any T. The result in para 1 (c) is explicable if the sum of the gap-values at any t is naively regarded as proportional to the number of superconducting electrons (or CPs) at that t. An example: at t = 0.3195 the sum of the two gap values is 8.0879 in Scenario 1 and 6.3336 meV in Scenario 2. This implies that there are fewer CPs in the latter scenario (and therefore more left over electrons as carriers of heat) than in the former. This is reflected in the κ_{s} values: 21.174 (Scenario 1), 26.409 W∙cm^{−1}∙deg^{−1} (Scenario 2).

3) The result in para 1 (b) implies that number of carriers in the stated range is more or less the same in both the scenarios; this however is not seen as convincingly as in the preceding case.

4) A feature of our approach is that we have not relied upon the Wiedemann-Franz law to separate out the lattice part of the total thermal conductivity. There is a difference of opinion about the utility of this law for the problem addressed here: while B et al. [_{2} at low temperatures.

5) We now report our findings about the data of S et al. [_{s}(T) reported by S et al. are lower than the values reported by B et al. [^{−1}deg^{−1} as against κ (38.7056) = 145.2463 mWcm^{−1}deg^{−1} quoted by B et al. We also note that B et al. have reported their data up to T = 7.4187 K, whereas S et al. have done so up to T = 1.8752 K. The differences in the values of κ_{s}(T) of the two groups can be ascribed to the compositional differences of the samples and the manners of their preparation since these can cause the scattering mechanism in the two samples to differ. This is an observation about which S has remarked [_{2} sample quality, I fully agree with you. It was the early time of superconducting MgB_{2}. Thus, sample composition, granularity etc. may differ significantly from group to group. This assumption is also supported by varying reported values of, e.g., the thermoelectric power.”

6) It seems interesting to point out that the approach followed in this paper is remarkably similar to the qualitative approach of Sologubenko et al. [_{1} and ∆_{2}, different parameters E_{1} and E_{2} of phonon-electron scattering, and separate contributions κ_{e}_{1} and κ_{e}_{2} to the heat transport.” In this manner—in the early days of MgB_{2}—they were able to give good estimates of the zero-temperature values of the two gaps.

1) We have presented here a detailed study of the thermal conductivity of MgB_{2} in the superconducting state via GBCSEs that were derived from a BSE by appealing to the twin concepts of a superpropagator and multiple Debye temperatures. This approach manifestly gives the ideas of Liu et al. [

2) A remarkable result of this paper is: while over a substantial range of temperatures below T_{c} the results in Scenario 1 are indistinguishable from those of Scenario 2, the latter scenario yields results in progressively better agreement with experiment when T ≤ 15 K. As is well known, thermal conductivity is a non-equilibrium phenomenon; it is measured under conditions of no electric current. Since a thermal current tends to drag a small electric current with it, this current must be balanced by an equal and opposite current. In an SC, it is balanced by a supercurrent. For these reasons, measurement of thermal conductivity requires a rather elaborate experimental setting up. It is not inconceivable therefore that cumulative effect of the stresses caused by this setting up lifts the “degeneracy” of the two gaps closing at the same T.

3) We finally note that the approach followed here may also shed light on whether or not the Tland the Bi-based HTSCs are characterized by three gaps—the possibility of which was suggested in [

G. P. M. thanks Prof. V. Z. Kresin for clarifying some points related to the original literature [4,5] on the subject matter of this paper. He also thanks Prof. D. C. Mattis for advice whenever approached. The authors are grateful to Prof. E. Bauer and Prof. M. Schneider for making their detailed experimental data available to them.