We propose that phonons can intrinsically 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 find that surface superconductivity dominates over bulk superconductivity in a range of chemical potentials around the Weyl nodes. The superconducting gap function realizes spinless chiral $p$-wave Cooper pairing in the Fermi arcs, leading to Majorana bound states in the core of vortices. Furthermore, we find a suppression of the absolute value of the gap in the center of the Fermi arcs, 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.

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• Electron-phonon coupling
• Fermi surface
• Majorana bound states
• Phonons
• Superconductivity
• Topological quantum computing
• Topological superconductors
• Topological materials
• Unconventional superconductors
• Weyl semimetal
• BCS theory
• Topology