On the Isotope-Like Effect for High-Tc Superconductors in the Scenario of 2-Phonon Exchange Mechanism for Pairing

By generalizing the isotope effect for elemental superconductors (SCs) to the case of pairing in the 2-phonon exchange mechanism for composite SCs, we give here an explanation of the well-known increase in the critical temperature (Tc) of Bi2Sr2CaCu2O8 from 95 K to 110 K and of Bi2Sr2Ca2Cu3O10 from 105 to 115 125 K when Bi and Sr in these are replaced by Tl and Ba, respectively. On this basis, we also give the estimated Tcs of some hypothetical SCs, assuming that they may be fabricated by substitutions similar to Bi → Tl and Sr → Ba.


Introduction
In this note we deal with an explanation of why the replacements of Bi and Sr in Bi 2 Sr 2 CaCu 2 O 8 and Bi 2 Sr 2 Ca 2 Cu 3 O 10 by Tl and Ba, respectively, lead to an increase in the critical temperature T c of the former from 95 K to 110 K and of the latter from 105 to 115 -125 K. This is an important undertaking because it has the potential to act as a general guide about substituting one or the other element in a composite superconductor (SC) in order to enhance its T c .
Empirically, it has been shown [1] that greater the T c of an SC, greater is its critical current density j 0 , for which, theoretically, an expression has been derived in terms of the following five parameters [2] [3]: Debye temperature θ, the How to cite this paper: Malik where k is the Boltzmann constant, P 0 the critical momentum of Cooper pairs and m* the effective mass of an electron. Therefore, if theory could predict the values of these parameters after one or more substitutions are made in an SC, then we would have a handle on its j 0 and T c . Since, as of now, theory cannot perform this task, we must resort to an approach that relies on a property or properties that can be unequivocally determined, regardless of the number of elements that are replaced by others.
Such a property is the mass of the SC.
Above considerations lead us to recall the isotope effect

Isotope-Like Effect in the 2 PEM Scenario
GBCSE for the T c of a composite SC in the 2 PEM scenario is [5] ( where chemical potential μ has been used interchangeably with E F , θ 1 and θ 2 are the Debye temperatures and λ 1 and λ 2 the interaction parameters of the ion-species responsible for pairing, and the operator Re ensures that the integrals yield real values even when ξ + μ < 0. When either of the λs is zero and μ >> kθ 1 (or kθ 2 ), (2) reduces to the usual BCS equation for the T c of an elemental SC in the 1 PEM scenario. Since T c of an SC in the 2 PEM scenario is due to the cooperative effect of two kinds of ions, the following generalization of (1) suggests itself naturally ( ) where p is the constant of proportionality and M 1 and M 2 are the masses of the ion-species that cause pairing. A discussion of (3) vis-à-vis (1) will be given below.

Bi2Sr2CaCu2O8
Pairing in this SC may be caused by the cooperative effect of one or more of the

Tl2Ba2CaCu2O8
We calculate T c (

Bi2Sr2Ca2Cu3O10 and Tl2Ba2Ca2Cu3O10
Following the same procedure as above, we can find 15 {α, p}-values, each of them corresponding to T c (Bi 2 Sr 2 Ca 2 Cu 3 O 10 ) = 105 K. Among these, the follow-

Discussion and Conclusion
It was noted above that BCS theory gives the value of α in (1) as 0.5. This follows from two relations: (a) ( ) ( ) where ω c is Debye frequency of the ions, N(0) the density of states at the Fermi surface, and V the net attraction between electrons bound as pairs, and (b) It is hence World Journal of Condensed Matter Physics seen that α = 0.5 only if N(0) and V do not change when one isotope is replaced by another. This is a reasonable assumption for N(0) because it is a purely electronic property; not so for V which is determined jointly by the ions and the electrons. It is not surprising therefore that α = 0.5 holds only for a few so-called classic elemental SCs, e.g., Zn, Pb, and Hg and that most of the other SCs are characterized by a multitude of values-some of which were noted above. Since, unlike elemental SCs, we do not have an analytic expression for the T c of a composite SC, we cannot derive for it a "blanket relation" such as α = 0.5. The value of α for such SCs is expected to differ from family-to-family and we believe to have indicated how it may be tested; besides, in the best-case scenario, it may prove to be useful in the current endeavor to reach room temperature T c s. when Bi and Sr in these are replaced by Tl and Ba, respectively. Based on this approach, we have given plausible values of T c s of some hypothetical SCs that may be obtained from the parent SCs by one or more substitutions. Fabrication of these hypothetical SCs, e.g., Tl 2 Be 2 CaCu 2 O 8 and ensuring that pairing in it occurs predominantly via the Be and the Ca ions (which lead to T c = 208 K), is a problem that belongs to the realm of chemical engineering.