High $T_c$ superconductivity at the FeSe/SrTiO$_3$ Interface

In a recent experiment the superconducting gap of a single unit cell thick FeSe film on SrTiO$_3$ substrate is observed by scanning tunneling spectroscopy and angle-resolved photoemission spectroscopy. The value of the superconducting gap is much larger than that of the bulk FeSe under ambient pressure. In this paper we study the effects of screening due to the ferroelectric phonons on Cooper pairing. We conclude it can significantly enhance the energy scale of Cooper pairing and even change the pairing symmetry. Our results also raise some concerns on whether phonons can be completely ignored for bulk iron-based superconductors.