Elastic and inelastic scattering in SrTiO$_{\mathrm{3-\delta }}$

Scattering among electrons generates a distinct contribution to electrical resistivity that follows a quadratic temperature dependence. We show that the prefactor of this T$^{\mathrm{2}}$ resistivity can be tuned by four orders of magnitude in metallic SrTiO$_{\mathrm{3}}$ by tuning the concentration of the carriers and consequently, the Fermi energy. The T$^{\mathrm{2}}$ behavior persists in the single-band dilute limit despite the absence of two known mechanisms for T$^{\mathrm{2}}$ behavior, distinct electron reservoirs and Umklapp processes. The ultimate origin of the small residual resistivity is the long Bohr radius, which, in a shallow Fermi sea caused by a random distribution of dopants, sets the zero-temperature mobility.