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SrTiO$_{3}$ undergoes a cubic-to-tetragonal phase transition at 105K. Thisantiferrodistortive transition is believed to be in competition with incipientferroelectricity. Substituting strontium by isovalent calcium induces aferroelectric order. Introducing mobile electrons to the system by chemicalnon-isovalent doping, on the other hand, leads to the emergence of a dilutemetal with a superconducting ground state. The link between superconductivityand the other two instabilities is an open question, which gathers momentum inthe context of the growing popularity of the paradigm linking unconventionalsuperconductors and quantum critical points. We present a set of specific-heat,neutron-scattering and dielectric permittivity and polarization measurements onSr$_{1-x}$Ca$_{x}$TiO$_{3}$ ($0<x<0.009$) and a low-temperature electricconductivity in Sr$_{0.9978}$Ca$_{0.0022}$TiO$_{3-\delta}$. Calciumsubstitution was found to enhance the transition temperature for bothanti-ferrodistortive and ferroelectric transitions. Moreover, we find thatSr$_{0.9978}$Ca$_{0.0022}$TiO$_{3-\delta}$ has a superconducting ground state.The critical temperature in this rare case of a superconductor with aferroelectric parent, is slightly lower than in SrTiO$_{3-\delta}$ ofcomparable carrier concentration. A three-dimensional phase diagram forSr$_{1-x}$Ca$_{x}$TiO$_{3-\delta}$ tracking the three transition temperaturesas a function of x and $\delta$ results from this study, in which ferroelectricand superconducting ground states are not immediate neighbours.

Interplay between antiferrodistortive, ferroelectric and superconducting instabilities in Sr_{1-x}Ca_{x}$TiO_{3-\delta}