Nonmonotonic anisotropy in charge conduction induced by antiferrodistortive transition in metallic ${\mathrm{SrTiO}}_3$

Cubic ${\mathrm{SrTiO}}_3$ becomes tetragonal below 105 K. The antiferrodistortive (AFD) distortion leads to clockwise and counterclockwise rotation of adjacent ${\mathrm{TiO}}_6$ 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 20% 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 ${\mathrm{SrTiO}}_3$.

Figure 1

(a) Left: Semilog plot of $\rho \left(T\right)$ in lightly doped ${\mathrm{SrTiO}}_{3-\delta }$ along two perpendicular axes. The inset is a zoom near the structural transition. No clear anomaly is visible at 105 K. Right: The ${\rho }_y/{\rho }_x$ ratio shows a clear upward deviation from a constant value, close to unity, at 105 K. The inset shows the geometry of the Montgomery method (see text). (b) Atomic configurations in the cubic and antiferrodistortive states. In the latter adjacent octahedra rotate clockwise and anticlockwise. (c) Representations of the three possible domains in the AFD state according to the orientation of the $c$ axis. Blue (red) squares represent octahedra which rotate clockwise (anticlockwise).

Figure 2

(a) and (b) The ${\rho }_y/{\rho }_x$ ratio as a function of $T$ in two ${\mathrm{SrTiO}}_{3-\delta }$ samples with different carrier densities under uniaxial strain. By applying strain along perpendicular orientations, the anisotropy is inverted. Curve indices correspond to a measurement sequence and the magnitude and orientation of strain. The $\epsilon =0$ data correspond to the strain-free configuration in the beginning and the end of one strain cycle. (c) Same for a Nb-doped sample. The inset in (a) shows the variation of the anisotropy as a function of strain at different temperatures.

Figure 3

(a)–(d) The evolution of the resistivity anisotropy anomaly in temperature for ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$. The anomaly shifts to higher temperatures with increasing doping. (e) Evolution of the AFD transition temperature with carrier concentration. The data reported in Ref. [22] are also included for comparison. Opposite trends are observed for alternative routes for doping. While Nb (or La) substitution enhances $T_{\text{AFD}}$, the introduction of oxygen vacancies depresses $T_{\text{AFD}}$.

Figure 4

(a) Theoretical Fermi surface in the cubic and the tetragonal states ($c/a=1.0006$). $E_{\text{F}}$ is placed 1 meV above the conduction band minimum. The Fermi surface is viewed along [001], [010], and [111]. (b) Band dispersion of STO for $c/a=1.0006$. (c) Fermi surface anisotropy (defined as the ratio of its radii along two perpendicular orientation) as a function of distortion, for different chemical potentials.

Figure 5

(a) ${\rho }_y$ and ${\rho }_x$ as a function of $T^2$ in strained ${\mathrm{SrTiO}}_{3-\delta }$. The anisotropy of the slope and the intercept are opposite, which points to a different anisotropy for elastic and inelastic components of resistivity. (b) Applying strain along $x$ favors the two strain domains whose longer $c$ axis is perpendicular to $x$. In the ($x,y$) plane, the twin boundary between these two domains is parallel to $x$ and scatters less electrons traveling along $x$, leading to a higher residual resistivity. On the other hand, the average Fermi velocity, set by the anisotropy of the Fermi surface, is larger along $x$. This leads to a lower inelastic resistivity for this orientation.

Figure 6

Theoretical Fermi surface anisotropy. Evolution of the Fermi surface (in the $a-c$ plane) with lattice distortion $c/a$ and chemical potential $E_F$. The distortion varies from 1 to 1.0006 and the values for chemical potential are 2, 3, 4, 5, and 10 meV.