Electron-Phonon Interaction and Superconductivity in In-Tl Alloys

The superconducting tunneling results obtained by Dynes for the electron-phonon part ${\alpha }^2\left(\omega \right)F\left(\omega \right)$ for indium-thallium alloys are examined in terms of both exact sum rules for alloy lattice dynamics and the self-consistent mass-defect theory. Detailed calculations are made of ${\alpha }^2\left(\omega \right)F\left(\omega \right)$ for four alloy concentrations. Good over-all agreement is found between calculated and tunneling-derived results. Our analysis seems to support Dynes's conclusion that the phase transition in the alloy structure near 30-at.% thallium could be due to phonon-mode softening. Zero-temperature Eliashberg equations are solved using the calculated and tunneling-derived values of ${\alpha }^2\left(\omega \right)F\left(\omega \right)$ to determine the gap and renormalization functions. Reduction in the dc Josephson current and condensation energies from their BCS value are calculated. Transition temperatures for the alloys are also calculated using the two sets of ${\alpha }^2\left(\omega \right)F\left(\omega \right)$. The agreement with the experiment is satisfactory.