We report the results of a calculation of the electron-phonon interaction in Y${\mathrm{Ba}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_7$ based on the nonorthogonal tight-binding approach to lattice dynamics. A self-consistent local-density electronic-structure calculation was combined with a shell-model description of the phonon spectrum. The resulting interaction matrix was used to calculate the superconducting transition temperature from a solution of the Eliashberg equations in which the full $\mathbf{k}$ dependence was retained. The transition temperature was found to be about 90 K, quite close to experiment. The oxygen isotope effect was investigated. We also calculated the $\mathbf{k}$-dependent gap function at $T=0\mathrm{}$. The gap shows significant variation between different sheets of the Fermi surface as well as dependence on $\mathbf{k}$ on the individual sheets, but is nodeless. The quantity $\frac{2\Delta }{k_B{T}_c}$ varies from 6.0 to 2.5 on the Fermi surface.

• Received 22 February 1994

DOI:https://doi.org/10.1103/PhysRevB.50.9511