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Non-monotonic anisotropy in charge conduction induced by antiferrodistortive transition in metallic SrTiO$_{3}$

Cubic SrTiO$_{3}$ becomes tetragonal below 105 K. The antiferrodistortive (AFD) distortion leads to clockwise and counter-clockwise rotation of adjacent TiO$_{6}$ octahedra. This insulator becomes a metal upon the introduction of extremely low concentration of n-type dopants. However, signatures of the structural phase transition in charge conduction have remained elusive. Employing the Montgomery technique, we succeed in resolving the anisotropy of charge conductivity induced by the AFD transition, in the presence of different types of dopants. We find that the slight lattice distortion ($<6 \times 10^{-4}$) gives rise to a twenty percent anisotropy in charge conductivity, in agreement with the expectations of band calculations. Application of uniaxial strain amplifies the detectable anisotropy by disfavoring one of the three possible tetragonal domains. In contrast with all other known anisotropic Fermi liquids, the anisotropy has opposite signs for elastic and inelastic scattering. Increasing the concentration of dopants leads to a drastic shift in the temperature of the AFD transition either upward or downward. The latter result puts strong constraints on any hypothetical role played by the AFD soft mode in the formation of Cooper pairs and the emergence of superconductivity in SrTiO$_3$.