Metallicity of oxygen-deficient EuTiO$_3$

We report on a study of charge transport in EuTiO$_{3-\delta}$ single crystals with carrier density tuned across several orders of magnitude. Comparing this system with other quasi-cubic perovskites, in particular strontium titanate, we draw a comprehensive picture of metal-insulator transition and dilute metallicity in this $AB$O$_3$ family. Because of a lower electric permittivity, the metal-insulator transition in EuTiO$_{3-\delta}$ occurs at higher carrier densities compared to SrTiO$_3$. At low temperature, a distinct $T^2$ resistivity is visible. Its prefactor $A$ smoothly decreases with increasing carrier concentration in a similar manner in three different perovskites. Our results draw a comprehensive picture of charge transport in doped quantum paraelectrics.