Superconductivity from orbital-selective electron-phonon coupling in AV3Sb5

Recent experiments have shown that the phase diagrams of the kagome superconductors $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ are strongly impacted by changes in the $c$-axis lattice parameter. Here, we show that $c$-axis deformations impact primarily the Sb apical bonds and thus the overlap between their ${p}_{z}$ orbitals. Changes in the latter, in turn, substantially affect low-energy electronic states with significant Sb character, most notably the central electron pocket and the van Hove singularities located above the Fermi level. Based on the orbital-selective character of $c$-axis strain, we argue that these electronic states experience a non-negligible attractive electron-phonon pairing interaction mediated by fluctuations in the apical Sb bonds. We thus propose a multiband model for superconductivity in $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ that includes both the Sb pocket and the V-derived van Hove singularities. Upon comparing the theoretical phase diagram with the experimentally observed vanishing of the ${T}_{c}$ dome across a Lifshitz transition of the Sb pocket, we propose that either an ${s}^{+\ensuremath{-}}$ or an ${s}^{++}$ state is realized in $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$.