  We investigate the origin of superconductivity in doped SrTiO$_3$ (STO) using
a combination of density functional and strong coupling theories within the
framework of quantum criticality. Our density functional calculations of the
ferroelectric soft mode frequency as a function of doping reveal a crossover
from quantum paraelectric to ferroelectric behavior at a doping level
coincident with the experimentally observed top of the superconducting dome.
Based on this finding, we explore a model in which the superconductivity in STO
is enabled by its proximity to the ferroelectric quantum critical point and the
soft mode fluctuations provide the pairing interaction on introduction of
carriers. Within our model, the low doping limit of the superconducting dome is
explained by the emergence of the Fermi surface, and the high doping limit by
departure from the quantum critical regime. We predict that the highest
critical temperature will increase and shift to lower carrier doping with
increasing $^{18}$O isotope substitution, a scenario that is experimentally
verifiable.
