Self-consistent surface superconductivity in time-reversal symmetric Weyl semimetals

Weyl semimetals host topologically protected surface states, the so-called Fermi arcs, that have a penetration depth into the bulk that depends on surface-momentum, and diverges at the Weyl points. It has recently been observed in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>PtBi</a:mi><a:mn>2</a:mn></a:msub></a:math> that such Fermi arc states can become superconducting, with a critical temperature larger than that of the bulk. Here we introduce a general variational method that captures the interplay between surface and bulk superconductivity for any bulk Hamiltonian that harbors (topological) surface states with varying penetration depth. From the self-consistent solutions, we establish that the surface state localization length of Weyl semimetals leads to characteristic features in the surface superconductivity, with a gap depending on surface momentum and a penetration length for the order parameter that is temperature-dependent due to competition with the bulk superconductivity.