Strong electron-phonon interaction in an FeSe monolayer*

We show that the electron-phonon coupling in an FeSe monolayer on a SrTiO$_3$ substrate is significantly larger than in earlier theoretical estimates. The role of the SrTiO$_3$ substrate is two-fold. First, the interaction of the FeSe and TiO$_2$ terminated face of SrTiO$_3$ prevents the FeSe monolayer from undergoing a shearing-type (orthorhombic) structural phase transition. Second, the substrate allows an anti-ferromagnetic ground state of FeSe which opens certain electron-phonon coupling channels within the monolayer that are prevented by symmetry in the non-magnetic phase. The spectral function for the electron-phonon coupling ($\alpha^2F$) in our calculations agrees well with inelastic tunneling data.