_{1}

The recent concern with the role of Fermi energy (
*E*
* _{F}*) as a determinant of the properties of a superconductor (SC) led us to present new

*E*

*-dependent equations for the effective mass (m*) of superconducting electrons, their critical velocity, number density, and critical current density, and also the results of the calculations of these parameters for six SCs the*

_{F}*T*

_{c}

*s*of which vary between 3.72 and 110 K. While this work was based on, besides an idea due to Pines, equations for Tc and the gap at

*T*= 0 that are explicitly

*E*

*-dependent, it employed an equation for the dimensionless construct that depends on*

_{F}*E*

*only implicitly;*

_{F}*k*in this equation is the Boltzmann constant, θ is the Debye temperature, and P0 is the critical momentum of Cooper pairs. To meet the demand of consistency, we give here derivation of an equation for y that is also explicitly

*E*

_{F}-dependent. The resulting framework is employed to (a) review the previous results for the six SCs noted above and (b) carry out a study of NbN which is the simplest composite SC that can shed further light on our approach. The study of NbN is woven around the primary data of Semenov et al. For the additional required inputs, we appeal to the empirical data of Roedhammer et al. and of Antonova et al.

Some of the recent studies [_{c} superconductors (SCs) have been motivated by the belief that Fermi energy (E_{F}) plays an important role in determining their T_{c}s and gap-structures. These studies make it natural to ask: why not incorporate E_{F} (equivalently, chemical potential μ) into the equations for the T_{c} and the gap _{F} because of the assumption

where k is the Boltzmann constant and

The proposed approach requires, besides the values of T_{c} and_{F}. Upon choosing critical current density _{0} and the following of their properties: m*, _{s}, which denote, respectively, the effective mass of superconducting electrons, their critical velocity at which _{2}, YBCO, Bi-2212, and Tl-2212 were also reported in [_{c}, an equation for the dimensionless construct_{F} only implicitly.

where m*, P_{0}, and E_{F} are in units of electron volts.

To meet the demand of consistency, we present here the derivation of a new equation for _{F} explicitly―to put it on par with the equations for T_{c} and

(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 _{e} 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 _{e} were not available. We were then led to estimate the values of _{c} and n_{e} for NbN, we can now also shed light on the ratio _{c}.

(ii) Since the value of the highest T_{c} reported for it in [_{2} for which, given its T_{c}, 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 θ_{NbN} into θ_{Nb} and θ_{N}.

The paper is organized as follows. In Section 2 are reproduced from [_{c} of an SC can be accounted for by a value of the interaction parameter

Recalled below from [_{F}, and _{c}, which is in accord with a tenet of the BCS theory. In the following we use _{F} interchangeably because they will be seen to differ negligibly. The modified equation for

Equation for

where

and

Equation for T_{c}:

where

and

In the above equations

After _{c}, and any assumed value of_{F} can be determined by the following equation

Equation for y:

This equation has been obtained by assuming that

where

Equation for j_{0}(E_{F}):

where

Equation (11) has been derived in [

In this equation

Equation (16) was obtained via a Bethe-Salpeter equation. It seems interesting to point out that when

The equation for the critical momentum

where

and we have used (9), (13) and (19). Besides, justification to follow, we have dropped E_{3} everywhere except in the denominator of (25) in order to avoid the singularity at

In order to obtain the

Because the constituents of both

where we have used (25), put

Therefore, for

Taking into account the overall sign of

where

Then substituting

where

and

obtain. It generalizes (11) which was obtained without this factor. While we could earlier solve (11) in the OPEM scenario with the input of _{F}. In order to carry out a quick consistency check of (32), we recall that upon solving (6) for Sn (θ = 195 K, T_{c} = 3.72 K,

Working in the OPEM scenario, we

(A) Solve. (6) with the input of θ and T_{c} to determine

(B) Solve (32) to obtain the values of

(C) Calculate

As predictions, this process also yields the values of m*, n_{s}, and

Before we can proceed as above, we need to fix the Debye temperature of the ions that cause pairing in NbN, i.e., θ_{Nb}.

θ_{NbN} is not quoted in [

We now need to resolve θ_{NbN} into θ_{Nb} and θ_{N}, which must be different because masses of Nb and N ions are different. As in [

where _{NbN} as in (33), the solutions of these equations yield

the corresponding values for θ_{N} 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 θ_{Nb}.

In [_{c} varying between 9.87 and 15.25 K have been reported for 13 samples of NbN for which the values of ^{−2}, the values of

If we solve the usual BCS equation for T_{c} (i.e., the equation sans E_{F}) with θ = 105.7 (397.8 K) and T_{c} = 10.72 K, we obtain λ = 0.4142 (0.2682). These are precisely the values we obtain via (6) for the same values of T_{c} and θ and the additional input of μ (or E_{F}) = 100 kθ for each value of θ being considered. Note that

Having fixed the values of θ_{Nb} and T_{c}, 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 [_{c}s under consideration

For each of the three values of T_{c} and both the values of θ_{Nb} noted above, we carried out steps (A)-(C) noted in Section (4.1) for_{Nb} = 105.7 K for only those values of _{NbN} and hence θ_{Nb} does not change significantly with T_{c}―as is seen from the data in [_{Nb} as 105.7 K, we have shown that each subset of the _{Nb}, we observe that (i) (3) and (6) can be employed only for values of_{Nb}, the value of

10.72 | 3.81 | 2.59 | 2.61 | 4.032 | 2.06 |

14.02 | 11.49 | 1.26 | 3.20 | 4.297 | 2.38 |

15.17 | 13.38 | 1.26 | 3.41 | 4.389 | 2.31 |

10.72 | 1 | 9.11 | 9.19 | 0.4300 | 4.496 | 13.158 | 3.65 | 1.79 |

14.02 | 4 | 36.4 | 36.5 | 0.4670 | 3.653 | 15.926 | 11.4 | 2.41 |

15.17 | 4.75 | 43.3 | 43.3 | 0.4845 | 3.418 | 16.971 | 13.6 | 2.64 |

We now take up the results following from θ_{Nb} = 397.8 K. The least permissible value of μ corresponding to it, i.e., ^{2} for T_{c} = 10.72 K and ^{2} for T_{c} = 14.02 K. Since both these _{Nb} because we had to employ the least value of _{c} = 10.72 K. As a concrete example in support of this statement, we note that θ_{Nb} = 125 K, led via the least permissible value of

Since this value of

Our considerations so far have been based on the derived values of θ_{Nb} from θ_{NbN} = 335 K. In order to find if there is a lower limit on the value of θ_{Nb}, we now report our findings based on the values of θ_{Nb} derived from the lowest value of θ_{NbN} that was noted above, i.e., 250 K. This value leads via (34) and (35) to θ_{Nb} = 296.8 (Nb as the upper bob) and θ_{Nb} = 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 _{Nb} cannot be as low as 78.9 K. The value closest to it that yields values of _{c}s is θ_{Nb} = 100 K, for which, e.g.,

Above considerations raise the question: Could

Given in

_{ } | |||||
---|---|---|---|---|---|

10.72 | 18.2 | 3.11 | 1.20 | 91.13 | 7.35 |

14.02 | 11.6 | 12.5 | 9.92 | 44.91 | 5.69 |

15.17 | 10.95 | 14.8 | 14.8 | 42.84 | 5.74 |

Notes: (i) The equations employed for the calculation of the above parameters have been derived in [_{s}(E_{F}) e v_{0}] at each T_{c} yields the same value for j_{0} as was calculated via (14) and given in _{c} = 10.72, 14.02 and 15.17 K, respectively, which justify the approximation made in obtaining (32).

the experimental values of T_{c} and _{c}s. Among these, the values of

For Sn and Pb, all our earlier results remain unchanged because solution of (32) for these elements yields the same values for

For each of the high-T_{c} SCs dealt with in [_{F}-dependent equation for

where _{2} was defined in (13). It is hence seen that

Equation (38) generalizes (32) to the TPEM scenario; because it explicitly contains E_{F} as a variable, it is also a generalized version of equation (30) in [_{F} as given in [_{c} SCs, we obtain the same value for

In connection with fixing θ_{Nb}, 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 [_{Nb}, for the identification of which we have simply employed (34) and (35) as a vehicle.

Among the five variables that determine

The main results of this paper are: (i) a new E_{F}-dependent equation for the dimensionless construct _{c}, _{Nb} in the range 100 - 106 K, (iii) predictions have been made about the values of m*, n_{s}, and _{c} and _{c}, and (v) it has been pointed out that we need to employ the new equations for

The work reported here is continuation of an attempt to find via theory tangible clues about raising the T_{c}s 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, T_{c}, _{e}, n_{s},

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 [

Malik, G.P. (2017) A Detailed Study of the Role of Fermi Energy in Determining Properties of Superconducting NbN. Journal of Modern Physics, 8, 99-109. http://dx.doi.org/10.4236/jmp.2017.81009