Quasi-isotropic orbital magnetoresistance in lightly doped ${\mathrm{SrTiO}}_3$

A magnetic field parallel to an electrical current does not produce a Lorentz force on the charge carriers. Therefore, orbital longitudinal magnetoresistance is unexpected. Here we report on the observation of a large and nonsaturating magnetoresistance in lightly doped $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3-x}$ independent of the relative orientation of current and magnetic field. We show that this quasi-isotropic magnetoresistance can be explained if the carrier mobility along all orientations smoothly decreases with magnetic field. This anomalous regime is restricted to low concentrations when the dipolar correlation length is shorter than the distance between carriers. We identify cyclotron motion of electrons in a potential landscape tailored by polar domains as the cradle of quasi-isotropic orbital magnetoresistance. The result emerges as a challenge to theory and may be a generic feature of lightly doped quantum paralectric materials.

Figure 1

Electrical transport properties of a lightly doped $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3-x}$ at low temperature : (a) Fermi surface of the lower band of $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ according to [27]. (b) Temperature dependence of the resistivity ($\rho$) of sample ${\mathrm{S}}_7$. Inset: $\rho$ vs $T^2$. (c) Transverse ($j\perp B$) and longitudinal ($j\parallel B$) magnetoresistance ($\frac{\mathrm{\Delta }\rho }{{\rho }_0}$=$\frac{\rho \left(B\right)-\rho (B=0)}{\rho (B=0)}$) as a function of the magnetic field up to 54 T, at $T=1.5$ K, for ${\mathrm{S}}_6$ $[n_H\left({\mathrm{S}}_6\right)=3.2\times {10}^{17}{\mathrm{cm}}^{-3}]$. (d) Normalized angular magnetoresistance of $S_1$ $[n_H\left(S_1\right)=6.5\times {10}^{16}{\mathrm{cm}}^{-3}]$ in polar plot at $B=10$ T for two temperatures: $T=2$ K (in blue) and $T=10$ K (in red). $\theta =0^{\circ }$ and ${90}^{\circ }$ correspond, respectively, to $\mathbf{j}\perp \mathbf{B}$ and $\mathbf{j}\parallel \mathbf{B}$).

Figure 2

Temperature dependence of the transverse and longitudinal magnetoresistance: (a)–(c) Field dependence of the resistivity for $\mathbf{j}\perp \mathbf{B}$ (${\rho }_{\perp }$), $\frac{{\rho }_{xy}}{B}$ and the resistivity for $\mathbf{j}\parallel \mathbf{B}$ (${\rho }_{\parallel }$) from $T=2$ to 14 K for $S_7$ $[n_H\left(S_7\right)=3.3\times {10}^{17}{\mathrm{cm}}^{-3}]$. (d)–(f) same as (a)–(c) from $T=16$ to 60 K. (g) Field dependence of the Hall mobility ${\mu }_H=\frac{{\rho }_{xy}}{B{\rho }_{\perp }}$ for $T=2$, 18, and 60 K. (h) Longitudinal magnetoconductivity ($\frac{1}{{\rho }_{\parallel }}$) compared with ${\sigma }_{\parallel }=ne{\mu }_H\left(B\right)$ with the deduced ${\mu }_H\left(B\right)$ shown in (g).

Figure 3

Doping dependence of the transverse and the longitudinal magnetoresistance at $T=2$ K: (a) and (b) TMR for $n_H$ ranging from $6.5\times {10}^{16}$ to $3.6\times {10}^{19}{\mathrm{cm}}^{-3}$. (c) and (d) same as (a) and (b) for the LMR. (e) Field dependence of Hall mobility (${\mu }_H$) deduced from the TMR and the Hall effect measurements in four low doped samples. (f) Comparison of the longitudinal magnetoconductance ($\frac{1}{{\rho }_{\parallel }}$) with ${\sigma }_{\parallel }=ne{\mu }_H\left(B\right)$ with the deduced ${\mu }_H\left(B\right)$ shown in (e).

Figure 4

Size of polar domains vs interelectron distance in lightly doped $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$: (a) Doping dependence of the ratio of the LMR and TMR at $B=10T$ for $T=2.5$ K (in blue closed circles). Inset: Sketch of the cubic unit cell of $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ lattices in the presence of an oxygen vacancy. (b) Doping dependence of ${\ell }_{ee}$=${\left(n_H\right)}^{-1/3}$ (the interelectron distance), of the screening length scale from charged impurities $r_{TF}=\sqrt{\frac{\pi a_B}{4k_F}}$ compares with the polar domain diameter, $2R_c=5.4$ nm at low temperature. (c) Sketch of the $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ cubic lattice in the presence of two oxygen vacancies separated by a distance ${\ell }_{ee}$ and of the polar domains (in light gray) which form around each oxygen vacancy with a radius $R_c$. Below a critical doping where ${\ell }_{ee}$ becomes shorter than $2R_c$ nonzero LMR appears.