Superconducting scanning tunneling microscopy tips in a magnetic field: Geometry-controlled order of the phase transition

The properties of geometrically confined superconductors significantly differ from their bulk counterparts. Here, we demonstrate the geometrical impact for superconducting scanning tunneling microscopy (STM) tips, where the confinement ranges from the atomic to the mesoscopic scale. To this end, we compare the experimentally determined magnetic field dependence for several vanadium tips to microscopic calculations based on the Usadel equation. For our theoretical model of a superconducting cone, we find a direct correlation between the geometry and the order of the superconducting phase transition. Increasing the opening angle of the cone changes the phase transition from first to second order. Comparing our experimental findings to the theory reveals first and second order quantum phase transitions in the vanadium STM tips. In addition, the theory also explains experimentally observed broadening effects by the specific tip geometry.