Phonon-Mediated Intrinsic Topological Superconductivity in Fermi Arcs

We propose that phonons can mediate topological superconductivity on the surface of Weyl semimetals. Weyl semimetals are gapless topological materials with nondegenerate zero energy surface states known as Fermi arcs. We derive the phonon spectrum and electron-phonon coupling in an effective model of a Weyl semimetal and apply weak-coupling Bardeen-Cooper-Schrieffer theory of superconductivity. In a slab geometry, we demonstrate that surface superconductivity dominates over bulk superconductivity in a wide range of chemical potentials. The superconducting gap function realizes spinless chiral $p$-wave superconductivity in the Fermi arcs, leading to Majorana bound states in the core of vortices. Furthermore, we show a suppression of the absolute value of the gap in the center of the arc, which is not captured by a local Hubbard attraction. The suppression is due to the nonlocal origin of electron-phonon coupling, leading to a layer dependence which has important consequences for topological surface states.