Quantum critical origin of the superconducting dome in SrTiO$_3$

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.