Vertex dominated superconductivity in intercalated FeSe

Bulk FeSe becomes superconducting below 9K but the critical temperature (Tc) is enhanced almost universally by a factor of ~4-5 when it is intercalated with alkali elements. How intercalation modifies the structure is known from in-situ X-ray and neutron scattering techniques, but there no consensus why Tc changes so dramatically. Here we show that there is one-to-one correspondence between the enhancement in magnetic instabilities and superconducting pairing vertex. Intercalation modifies electronic screening both in the plane and also between layers. We disentangle quantitatively how superconducting pairing vertex gains from each such changes in electronic screening. Intercalated FeSe provides an archetypal example of superconductivity where information derived from the single-particle electronic structure is not sufficient to account for the origins of superconductivity, even when they are computed including correlation effects. Finally, we show that, whether intercalated or not, Tc is not sensitive to the exact position of the dxy level at the Gamma point, as dxy character is always present in the electron pockets and can mediate similar Tc enhancements.