Some of the recent studies  -  concerned with high-Tc superconductors (SCs) have been motivated by the belief that Fermi energy (EF) plays an important role in determining their Tcs and gap-structures. These studies make it natural to ask: why not incorporate EF (equivalently, chemical potential μ) into the equations for the Tc and the gap of an SC, and then treat it as an independent variable? This is a departure from the usual practice because these parameters are conventionally calculated via equations sans EF because of the assumption
where k is the Boltzmann constant and is the Debye temperature.
The proposed approach requires, besides the values of Tc and, another property of the SC in order to determine EF. Upon choosing critical current density of the SC, new equations for both elemental and composite SCs valid at T = 0 were recently presented in  for j0 and the following of their properties: m*, , and ns, which denote, respectively, the effective mass of superconducting electrons, their critical velocity at which vanishes, and the density of superconducting electrons. While the results of such a study for Sn, Pb, MgB2, YBCO, Bi-2212, and Tl-2212 were also reported in  , it was based on, unlike the equations for and Tc, an equation for the dimensionless construct, defined below, that is dependent on EF only implicitly.
where m*, P0, and EF are in units of electron volts.
To meet the demand of consistency, we present here the derivation of a new equation for that also contains EF explicitly―to put it on par with the equations for Tc and. While this leads us to review our earlier results, we also undertake here a detailed study of the superconducting properties of NbN because:
(i) It is the simplest composite SC different samples of which (a) have been fabricated by the same method of preparation, (b) are geometrically similar, but (c) differ in size (e.g., film thickness), and for which (d) data in the form are available, where ne is the density of conduction electrons. This is unlike the composite SCs dealt with earlier, which were not necessarily fabricated by the same method of preparation and for which the values of and ne were not available. We were then led to estimate the values of for these SCs from the data at T = 4.2 K. Given the values of Tc and ne for NbN, we can now also shed light on the ratio as a function of Tc.
(ii) Since the value of the highest Tc reported for it in  is 15.25 K, it is the simplest composite SC for which we believe one-phonon exchange mechanism (OPEM) to be operative. This is unlike, e.g., MgB2 for which, given its Tc, we need to invoke the two- phonon exchange mechanism (TPEM).
(iii) The above features make NbN the simplest testing ground for some key steps of our approach, such as the procedure followed for resolving θNbN into θNb and θN.
The paper is organized as follows. In Section 2 are reproduced from  those equations that constitute our framework in the OPEM scenario, which may be defined as one in which the Tc of an SC can be accounted for by a value of the interaction parameter that satisfies the Bogoliubov constraint, i.e., λ < 0.5. Section 3 is devoted to derivation of the new equation for. The study of NbN is taken up in Section 4. A review of our earlier results is taken up in Section 5. The final two sections are devoted to a discussion and conclusions, respectively.
2. EF-Incorporated Equations for Various Properties of an SC
Recalled below from  are some of the equations that we need for NbN. In these equations is to be identified with. Further, the equations have been written by assuming that, EF, and have the same values at T = 0 and T = Tc, which is in accord with a tenet of the BCS theory. In the following we use and EF interchangeably because they will be seen to differ negligibly. The modified equation for will be derived in the next section.
Equation for Tc:
In the above equations
After has been determined via (7) with the input of θ, Tc, and any assumed value of, the corresponding value of EF can be determined by the following equation
Equation for y:
This equation has been obtained by assuming that
Equation for j0(EF):
3. The Modified Equation for y in the OPEM Scenario
Equation (11) has been derived in  (pp. 115-120) by assuming Inequality (1). In order to do away with this inequality, we begin here with the following equation for moving CPs because the present derivation differs from the earlier one only beyond it.
In this equation
Equation (16) was obtained via a Bethe-Salpeter equation. It seems interesting to point out that when, it reduces to the well known criterion of superconductivity derived by Thouless via the t-matrix approach, as can be seen from  and, in greater detail, in  .
The equation for the critical momentum at any temperature follows from (16) by putting W = 0. In terms of we then have
and we have used (9), (13) and (19). Besides, justification to follow, we have dropped E3 everywhere except in the denominator of (25) in order to avoid the singularity at. Compared with the earlier equation for, the new feature of (22) is that it has the additional factor of in each of its constituents.
In order to obtain the version of (22), we split both and into two parts: into and for which the limits of integration are and respectively, and into and, where the former is integrated from and the latter from It is then seen that, when T = 0, for and and (+1) for the remaining parts.
Because the constituents of both and differ from one another only in the matter of limits and an overall sign, we now consider the following indefinite integral:
where we have used (25), put and whence
Therefore, for (as will be seen to be so), we obtain
Taking into account the overall sign of, (28) yields
where Since we replace in the above equation by in order to make contact with (11)., and can be similarly calculated. For the sake of compactness, we define
Then substituting and into (22), we obtain
and has been defined for later convenience. Obtained by retaining the factor in and, (32) for y is the equation we had set out to
obtain. It generalizes (11) which was obtained without this factor. While we could earlier solve (11) in the OPEM scenario with the input of alone, solution of (32) requires the additional input of θ and EF. In order to carry out a quick consistency check of (32), we recall that upon solving (6) for Sn (θ = 195 K, Tc = 3.72 K,), we had earlier obtained λ = 0.2466. The solution of (11) then led to. This is precisely the value we now obtain by solving (32) with the same inputs for, , and
4. Study of NbN Based on EF-Incorporated Equations
4.1. Outline of Procedure
Working in the OPEM scenario, we
(A) Solve. (6) with the input of θ and Tc to determine for different assumed values of.
(B) Solve (32) to obtain the values of corresponding to each pair of values obtained above.
(C) Calculate via (14) for each triplet of values till it is found to agree with its experimental value.
As predictions, this process also yields the values of m*, ns, and via equations derived in  and noted in Table 3. As a further check, we calculate via (3) by employing the values of and that led in (C) to the experimental value of.
Before we can proceed as above, we need to fix the Debye temperature of the ions that cause pairing in NbN, i.e., θNb.
4.2. Debye Temperature of Nb Ions in NbN
θNbN is not quoted in  . The reported values for it vary in the range 250 - 335 K     . We begin by adopting 
We now need to resolve θNbN into θNb and θN, which must be different because masses of Nb and N ions are different. As in  , we do so via the following equations
where () is the atomic mass of N (Nb). While the first of the above equations has been routinely used for binaries, the second equation has been derived  by assuming that the constituents of the binary simulate weakly coupled oscillations of a double pendulum. The equations above have been written by assuming that Nb is the upper bob of the double pendulum. With, , and θNbN as in (33), the solutions of these equations yield
the corresponding values for θN being 272.2 and 564.3 K (which we do not need). In the following we shall perform all calculations with both the above values of θNb.
4.3. Choosing the Values of Tc for Which the Data in  Are Addressed
In  , while values of Tc varying between 9.87 and 15.25 K have been reported for 13 samples of NbN for which the values of lie in range 2.92 - 13.30 MA∙cm−2, the values of have been reported at only three values of, which are 10.72, 14.02, and 15.17 K. Hence we limit the scope of this paper to these values of only.
4.4. A Consistency Check of (6)
If we solve the usual BCS equation for Tc (i.e., the equation sans EF) with θ = 105.7 (397.8 K) and Tc = 10.72 K, we obtain λ = 0.4142 (0.2682). These are precisely the values we obtain via (6) for the same values of Tc and θ and the additional input of μ (or EF) = 100 kθ for each value of θ being considered. Note that manifestly satisfies constraint (1). It is hence seen that (6) incorporating is a valid generalization of the usual equation sans, and may therefore be used for arbitrary values of.
4.5. Fixing Additional Required Inputs
Having fixed the values of θNb and Tc, we can carry out steps (A) and (B) spelled out in Section 4.1; to carry out step (C) we additionally need the values of γ and the cell parameters of different samples of NbN, which are not given in  . We fix these by appealing to the data in  . A summary of all the inputs required for this study is given in Table 1. Based on the data in  , this table includes the estimated values of at each of the Tcs under consideration
For each of the three values of Tc and both the values of θNb noted above, we carried out steps (A)-(C) noted in Section (4.1) for. For the sake of brevity, presented in Table 2 are the results corresponding to θNb = 105.7 K for only those values of for which the calculated values of are in close agreement with their experimental values noted in Table 1. In obtaining these results we have assumed that θNbN and hence θNb does not change significantly with Tc―as is seen from the data in  . Thus, up to this stage, having fixed the value of θNb as 105.7 K, we have shown that each subset of the experimental values can be accounted for by a corresponding set of values. Since it is pertinent to ask if we could have achieved similar agreement by adopting a different value of θNb, we observe that (i) (3) and (6) can be employed only for values of―otherwise we run into complex values because of the factor; (ii) for μ as any multiple of kθNb, the value of calculated via either of these equations must be less than 0.5 in order to satisfy the Bogoliubov constraint, and (iii) for any value of increases as μ is increased.
Table 1. Experimental values of  ,  , and  employed for the study of NbN in this paper.
Table 2. Results of calculations for θNb = 105.7 K. The value of against each Tc is the one that led―via the values of EF, , , and (the gram-atomic volume of NbN)―to a value of in close agreement with its experimental value noted in Table 1. was calculated with the input of a0 from Table 1 and the atomic masses of the Nb and N, as in  .
We now take up the results following from θNb = 397.8 K. The least permissible value of μ corresponding to it, i.e., led to MA/cm2 for Tc = 10.72 K and MA/cm2 for Tc = 14.02 K. Since both these values are greater than their experimental counterparts, in the light of observation (iii) above, one might attempt to employ lower values of―which is ruled out because of (i). In fact the value 105.7 K seems like the upper limit for θNb because we had to employ the least value of corresponding to it in order to achieve agreement between the calculated and the experimental values of at Tc = 10.72 K. As a concrete example in support of this statement, we note that θNb = 125 K, led via the least permissible value of corresponding to it to the following results:
Since this value of exceeds the experimental value, we need to employ a lower value of―which is impermissible because we have already employed for it the lowest allowed value.
Our considerations so far have been based on the derived values of θNb from θNbN = 335 K. In order to find if there is a lower limit on the value of θNb, we now report our findings based on the values of θNb derived from the lowest value of θNbN that was noted above, i.e., 250 K. This value leads via (34) and (35) to θNb = 296.8 (Nb as the upper bob) and θNb = 78.9 K (Nb as the lower bob). Since the former of these values exceeds the upper limit noted above, we did not pursue it any further. For the latter value, we obtained for any assumed value of
Because both these values of are in conflict with the Bogoliubov constraint, we conclude that θNb cannot be as low as 78.9 K. The value closest to it that yields values of satisfying the Bogoliubov constraint at both the Tcs is θNb = 100 K, for which, e.g.,
Above considerations raise the question: Could for NbN? If so, it would put NbN in the category of heavy-fermion SCs  . Since there is no compelling reason to believe that this may be so, we did not pursue this idea.
Given in Table 3 are the predicted values of various parameters concomitant with
Table 3. With θNb = 105.7 K, predicted values of various parameters of NbN that are concomitant with the calculated values of given against each Tc in Table 2.
Notes: (i) The equations employed for the calculation of the above parameters have been derived in  and are as follows: (ii) The product [ns(EF) e v0] at each Tc yields the same value for j0 as was calculated via (14) and given in Table 2. (ii) The values of are, and for Tc = 10.72, 14.02 and 15.17 K, respectively, which justify the approximation made in obtaining (32).
the experimental values of Tc and of NbN at the three Tcs. Among these, the values of are in reasonably good agreement with their experimental counterparts (see Table 1), considering that the latter are estimated values based on the data of  .
5. A Review of the Results Obtained in  in View of the Modified Equation for y
For Sn and Pb, all our earlier results remain unchanged because solution of (32) for these elements yields the same values for that were obtained via (11). Since the values of that were needed for these elements are rather large, 55 for Sn and 108 for Pb, this result was to be expected; it also establishes that (32) is a valid generalization of (11). To bring out the extent to which the solutions of the two equations differ for low values of, we note that if we erroneously employ (11) for Sn for, θ = 195 K and λ = 0.2516 (these values are consistent with of the SC), then we obtain y = 20.083  ; employment of (11) in this case is erroneous because the equation was obtained by assuming that. On the other hand, solution of (32) for this case leads to.
For each of the high-Tc SCs dealt with in  , there are two―say, and― and two in the problem. The EF-dependent equation for that we now need to employ is
where was defined in (32), θ being the Debye temperature of the SC, and E2 was defined in (13). It is hence seen that―as it ought to be. Without the multipliers and, would denote in the first term on the RHS of (38) and in second term, whereas with the multipliers has the same definition (i.e.,) for both the terms.
Equation (38) generalizes (32) to the TPEM scenario; because it explicitly contains EF as a variable, it is also a generalized version of equation (30) in  . Upon solving (38) with the input of, and EF as given in  for any of the high-Tc SCs, we obtain the same value for that we had obtained earlier. Notwithstanding the fact that all our results reported in  remain unchanged is fortuitous―for lower values of than were required in  , we ought to employ the more accurate (38) rather than equation (30) in  .
In connection with fixing θNb, we recall that Debye temperature is just another way to specify Debye frequency; it is not to be confused with thermodynamic temperature. We now note that, based on neutron powder diffraction experiments, different values of Debye temperature for the constituents of anisotropic LCO have been reported  . This lends support to the idea that the Debye temperature of a composite SC needs to be “resolved.” The results reported here depend only on the value of θNb, for the identification of which we have simply employed (34) and (35) as a vehicle.
Among the five variables that determine―see Equation (14) ―seems to stand alone. We draw attention to a discussion of this variable in  .
The main results of this paper are: (i) a new EF-dependent equation for the dimensionless construct defined in (2) has been derived, (ii) it has been shown that the experimental values of Tc, , and of NbN are explicable in the OPEM scenario by a value of θNb in the range 100 - 106 K, (iii) predictions have been made about the values of m*, ns, and that are concomitant with the Tc and values of NbN, (iv) the greater the value of the ratio, the greater is the value of Tc, and (v) it has been pointed out that we need to employ the new equations for presented here when.
The work reported here is continuation of an attempt to find via theory tangible clues about raising the Tcs of composite SCs. The role of experiment in this quest can hardly be over-emphasized. While huge amounts of such data about hundreds of SCs are now available, we have not come across a single composite SC for which all the relevant parameters identified here, i.e., θ, Tc, , , m*, , ne, ns, , and, have been reported.
We conclude by noting that the derivations of most of the equations employed in this paper and the concepts on which they are based, e.g., multiple Debye temperatures, superpropagator, and the Bogoliubov constraint, can be found at one place in  .
The author thanks Dr. A. Semenov for kindly responding to his queries concerned with the experimental data reported in  , and Professor D.C. Mattis for encouragement.