  The interface superconductivity in LaAlO$_{3}$-SrTiO$_{3}$ heterostructures
reveals a non-monotonic behavior of the critical temperature as a function of
the two-dimensional density of charge carriers. We develop a theoretical
description of interface superconductivity in strongly polar heterostructures,
based on the dielectric function formalism. The density dependence of the
critical temperature is calculated accounting for all phonon branches including
different types of optical (interface and half-space) and acoustic phonons. The
LO- and acoustic-phonon-mediated electron-electron interaction is shown to be
the dominating mechanism governing the superconducting phase transition in the
heterostructure.

Superconductivity