We perform first-principles calculations of the band and $\mathit{k}$-point resolved superconducting gap of Pb in the framework of the density functional theory for superconductors. Without any adjustable parameter or assuption different from $s$-wave symmetry, we find two different values of the gap on the two sheets of the Fermi surface, which can be related to the different electron-phonon couplings characterizing the electronic states in the corresponding bands. These, in turn, derive from the different orbital character of the electronic states. We also find some intraband gap anisotropy in each Fermi surface sheet. Our calculated gap, critical temperature and total anisotropy of the gap are in good agreement with tunneling experiments. We estimate an $\approx 8\%$ enhancement of $T_c$ coming from the gap anisotropy. However, the experimentally found $T^3$ temperature dependence of the specific heat cannot be found within our assumed anisotropic $s$-wave gap symmetry.

• Received 31 October 2006

DOI:https://doi.org/10.1103/PhysRevB.75.054508