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Recent experiments have shown that the phase diagrams of the kagomesuperconductors $A\mathrm{V}_3\mathrm{Sb}_5$ are strongly impacted by changesin the $c$-axis lattice parameter. Here, we show that $c$-axis deformationsimpact primarily the Sb apical bonds and thus the overlap between their $p_z$orbitals. Changes in the latter, in turn, substantially affect low-energyelectronic states with significant Sb character, most notably the centralelectron pocket and the van Hove singularities located above the Fermi level.Based on the orbital-selective character of $c$-axis strain, we argue thatthese electronic states experience a non-negligible attractive electron-phononpairing interaction mediated by fluctuations in the apical Sb bonds. We thuspropose a multi-band model for superconductivity in$A\mathrm{V}_3\mathrm{Sb}_5$ that includes both the Sb pocket and the V-derivedvan Hove singularities. Upon comparing the theoretical phase diagram with theexperimentally observed vanishing of the $T_c$ dome across a Lifshitztransition of the Sb pocket, we propose that either an $s^{+-}$ or an $s^{++}$state is realized in $A\mathrm{V}_3\mathrm{Sb}_5$.

Superconductivity from Orbital-Selective Electron-Phonon Coupling in $A\mathrm{V}_3\mathrm{Sb}_5$