Isotope effect in the translation-invariant bipolaron theory of high-temperature superconductivity

It is shown that the translation-invariant bipolaron theory of superconductivity explains the dependence of the isotope coefficient in high-temperature superconductors on the critical temperature of a superconducting transition: in the case of strong electron-phonon interaction the isotope coefficient is low when doping is optimal and high when it is weak. It is demonstrated that in the case of London penetration depth the absolute value of the isotope coefficient behaves in all the opposite way. A conclusion of the great role of non-adiabaticity in the case of weak doping is made.


Introduction.
The isotope coefficient plays the central role in superconductivity (SC). The occurrence of the isotope effect had been decisive in revealing the phonon mechanism of SC in ordinary superconductors. A lack of this effect in optimally doped high-temperature superconductors (HTSC) was the reason for discarding the phonon mechanism in HTSC and, as a consequence, the Bardeen-Cooper-Schrieffer theory (BCS) [1]. In recent years, however, a lot of new experimental facts has led researchers to return to the electron-phonon interaction (EPI) as a dominant mechanism for explaining the HTSC effect. At the same time the direct use of the BCS and its various modifications cannot explain these experimental facts [2].
The reason is probably that the BCS, being based on EPI, considers this interaction to be weak while in the case of HTSC it turns out to be strong. Generalization of the BCS to the case of strong EPI -Eliashberg theoryhas been unable to explain many important phenomena attending HTSC, such as the pseudogap state. To overcome these difficulties the author has developed a translation-invariant (TI) bipolaron theory of HTSC where the role of Cooper pairs belongs to TI bipolarons [3] - [6]. The aim of this paper is to explain isotope effects observed in HTSC on the basis of the TI bipolaron theory.

Isotope influence on
The isotope influence on the transition temperature played a decisive role in revealing the electron-phonon mechanism of a superconducting state and substantiating the Bardeen-Cooper-Schrieffer theory [1] for conventional superconductors. In the BCS theory, the isotope coefficient α for is determined from the relation found experimentally for ordinary metals, such that: where is the mass of an atom replaced by its isotope. It follows from (1) that: In the BCS theory, the value of α is positive and close to α ≅ 0,5, which is in good agreement with the experiment in ordinary metals. The great value of the isotope coefficient observed in ordinary metals implies a dominant role of EPI in them and suggests the applicability of the BCS theory for their description.
On the contrary, in high-temperature superconducting ceramics (HTSC), the isotope coefficient α is, generally, very small (~10 −2 ) in the region of their optimal doping which suggests that EPI there is negligible and other mechanisms of SC should be evoked [2].
As it is known, the BCS theory developed for the case of weak EPI is inapplicable in the case of HTSC where EPI is strong. In this case of use may be the translation-invariant bipolaron theory developed in [3][4][5][6] (the reasons for which the Eliashberg theory [7] which is used in the case of strong EPI, can be unsuitable to describe HTSC are discussed in [6]).
According to the TI bipolaron theory, the temperature of a SC transition is determined by the equation [3][4][5][6]: where is the concentration of TI bipolarons, 0 is the frequency of an optical phonon, mis the mass of a band electron (hole), ħ=h/2 , h -is the Plank constant.
The graph of the function α( /ω0) is given in Fig. 1 Fig . 1 Dependence of the isotope coefficient α on the value of y = / 0 .
The curve in Fig1 shows that as isotope coefficient decreases the transition temperature increases. Fig.1 also suggests that in the case of /ω0 ≫ 1, which can correspond to optimally doped HTSC, the isotope coefficient will be small (α → 0 for Tc / 0 → ∞), in full agreement with the experiment [2], [8]- [12].
In the opposite case: / 0 ≪ 1, which corresponds to small doping, the isotope coefficient is maximum and equal to α=0,5, as in the BCS theory.
Notice that according to (4) the isotope coefficients of different samples are the same for the same relations of / 0 . Figure 1 suggests that for typical values of α lying in the interval (0.25 ÷0.5) the value of phonon frequency 0 not exceed . For HTSC with = 100K this leads to 0 be less then 8.6 meV.

Isotope influence on London penetration depth.
Quite a different picture is observed for the isotope coefficient for London penetration depth : , c-is the velocity of light, e is the electron charge, * − is the bipolaron mass, 0 − is the concentration of TI bipolarons in Bose condensate 0 = 0 / , 0 is the number of TI bipolarons in the condensate [3]- [6]: N is the total number of bipolarons, = ( 2/3 2 ħ 2 * ) , ̃= / * , * is the scale multiplier with As noted in [6], the mass of a TI bipolaron does not differ greatly from 2m where m is the mass of a band electron which depends on 0 only slightly. For this reason we will believe that the whole dependence on 0 is determined by involved in (6) the concentration of TI bipolarons in the condensate 0 , related with 0 by formula (7). As a result, it follows from (5)-(7) for the isotope coefficient β: .
In the case of 0 / ≪ 1, 1/2 ( − 0/ ) = √ 0 , the isotope coefficient is equal to: It should be noted that as distinct from the coefficient α, which is positive, the for London penetration depth is negative, which is in agreement with the experiment. The fact that in the limit of low temperatures (9) the isotope coefficient β, caused by EPI is negligible is consistent with the BCS [2]. Hence, in the limit of weak doping (when → 0) the isotope coefficient for London penetration depth as distinct from the isotope coefficient α for (which is high in this case) will be very small. The experiment, however, shows that the value of in the case of optimal doping (when is maximum) in the limit of low temperatures can be very large. For example, in optimally doped HTSC 2 3 7 at =0 the value of | (0)| ≈ 0,2| [13]. Therefore the main contribution in this case can be made by a non-adiabatic mechanism or non-phonon mechanisms [2], [13]. An example of HTSC in which the contribution of non-adiabaticity and non-phonon mechanisms is probably small is provided by slightly overdoped 2− 1− 4 , for which at =0 the isotope coefficient in accordance with the theory developed vanishes [14].
At high temperatures, on the contrary, the main contribution near can provide a phonon mechanism determined by (10) (according to [15] a value α=0.025 as observed in 2 3 7− one obtains ~−0.6 for /~0.95).

Conclusion
The results obtained enable us to explain the peculiarities in the behavior of the isotope coefficient α observed for in high-temperature superconductors, in particular, its small value in the case of optimal doping and high value in the case of weak doping relying on merely the electron-phonon interaction. The problem of the isotope coefficient for London penetration depth λ and its temperature dependence is more complicated. The electron-phonon interaction explains high values of for optimally-doped HTSC materials only in the vicinity of the SC transition temperature . In the case of low temperatures the theory explains negligible values of (0)in HTSC 2− 1− 4 and does not explain high values of (0) in other HTSC materials. In papers [2], [13] this disagreement with the theory developed and the BCS is explained by the fact that in many HTSC materials at low temperatures the main role belongs to non-adiabaticity effects which lead to high values of (0).