Conditions for $T^{2}$ resistivity from electron-electron scattering

Many complex oxides (including titanates, nickelates and cuprates) exhibit a carrier scattering mechanism with a power law dependence on temperature (resistivity $\rho \propto T^{2})$. By analogy to a similar phenomenon observed in metals at low temperature, this mechanism has often been identified as Fermi-liquid-like electron-electron scattering (Baber scattering). However, other transport signatures in these materials have shown behavior that casts doubt on this simple picture. Careful investigation of electron-electron scattering (both direct and phonon-mediated) reveals that the Baber $T^{2}$ power law rests on several crucial assumptions. In some cases, these assumptions are not satisfied and removing them destroys the power law. We illustrate these issues with two case studies: sodium metal (in which electron-electron scattering gives $T^{2}$ resistivity) and strontium titanate (in which it does not). Our results suggest that an observation of $\rho \propto T^{2}$ is not sufficient evidence for electron-electron scattering. The power law observed in the complex oxides may instead be due to another, as yet undiscovered, mechanism.