Vortex bound states and Majorana zero mode from superconducting proximity effect

We study a thin metal film in proximity with an s-wave superconductor under a perpendicular magnetic field leading to Abrikosov vortices, when the coherence length in the proximitized layer is much larger than the one of the parent superconductor ξ_N >> ξ_S. The induced pair potential restores to its value at infinity over the distance ξ_S from the vortex center, rather than the induced coherence length ξ_N. Using quasiclassical approximation, we show that the presence of multiple lengthscales leads to qualitative changes in the properties of the vortex bound states in comparison to conventional ones.

When the London penetration depth is large λ >> ξ_N >> ξ_S, in the limit ξ_S→0 the energy spacing acquires a logarithmic correction in the angular momentum of the bound state. The wavefunctions are localized at the larger scale ξ_N, which will lead to observable consequences, such as quasiparticle interference pattern in the presence of defects.

Another hierarchy can also be realized: ξ_N >> λ >> ξ_S, and we show that the Majorana zero mode is robust the case of linearly dispersing bands in the proximitized layer.

Lastly, we put our results in the context of gapless superconductivity in proximitized layers that was recently realized experimentally [arXiv:2010.02216].