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Effect of thermal phonons on the superconducting transition temperature

1983; American Physical Society; Volume: 28; Issue: 3 Linguagem: Inglês

10.1103/physrevb.28.1304

1095-3795

C.R. Leavens, Etienne Talbot,

Magnetic and transport properties of perovskites and related materials

There is no consensus in the literature on whether or not thermal phonons depress the superconducting transition temperature ${T}_{c}$. In this paper it is shown by accurate numerical solution of the real-frequency Eliashberg equations for the pairing self-energy $\ensuremath{\varphi}$ and renormalization function $Z$ that thermal phonons in the kernel for $\ensuremath{\varphi}$ raise ${T}_{c}$ but those in $Z$ lower it by a larger amount so that the net effect is to depress ${T}_{c}$. (A previous calculation which ignored the effect of thermal phonons in $\ensuremath{\varphi}$ overestimated the suppression of ${T}_{c}$ by at least a factor of 3.) It is shown how to switch off the thermal phonons in the imaginary-frequency Eliashberg equations, exactly for $Z$ and approximately for $\ensuremath{\varphi}$. The real-frequency and approximate imaginary-frequency results for the depression of ${T}_{c}$ by thermal phonons are in very satisfactory agreement. Thermal phonons are found to depress the transition temperature of ${\mathrm{Nb}}_{3}$Sn by only 2%. It is estimated that the suppression of ${T}_{c}$ by thermal phonons saturates at about 50% in the limit of very strong electron-phonon coupling.