  Monolayer films of FeSe grown on SrTiO$_3$ substrates are electron doped
relative to bulk and exhibit a significantly higher superconducting transition
temperatures. We present density functional calculations and a modified
Schottky model incorporating the strong paraelectricity of SrTiO$_3$ which
demonstrate that the doping may be due to charge transfer from SrTiO$_3$
impurity bands driven by work function mismatch. Physically relevant levels of
Nb doping are shown to lead to doping of the FeSe compatible with observation.
The coupling of electrons in FeSe to polar phonons in the depletion regime of
the SrTiO$_3$ is calculated. A $\lambda\sim 0.4$ is found; the coupling to
long-wavelength phonons is found to be dominant.
