Weak localization and spin-orbit interaction in side-gate field effect devices at the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface

Using field effect devices with side gates, we modulate the 2-dimensional electron gas hosted at the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface to study the temperature and doping evolution of the magnetotransport. The analysis of the data reveals different transport regimes depending on the interplay between the different (elastic, inelastic, and spin-orbit) scattering times and their temperature dependencies. We find that the spin-orbit interaction strongly affects the low-temperature transport in the normal state in a very large region of the phase diagram, extending beyond the superconducting dome.

Figure 1

(a) Sketch of the device geometry showing the electrical connections used to realize measurements under field effect. The blue area indicates the 2DEG. (b) Zoom of the central area of the device: $w$ is the channel width and $t$ is the distance between each side gate and the channel edge. (c) Finite-element simulations of the electric field intensity: the $yz$ plane at the middle of the channel is shown. The electric field lines are drawn in white.

Figure 2

(a) Sheet resistance $R_s$ vs gate voltage $V_{sg}$ for two side-gate field effect devices having the same width $w=0.25$ $\mu \mathrm{m}$ and different distance $t$ from the side electrodes to the channel edge ($t=1.7$ $\mu \mathrm{m}$ for green data and 0.7 $\mu \mathrm{m}$ for orange data). (b) $R_s$ vs temperature T for three selected carrier concentration. The carrier concentration is indicated using the value of the sheet conductance ${\sigma }_{2\mathrm{D}}^0$, as explained in the text.

Figure 3

Magnetoconductance curves of a side-gate field effect device with $w=0.75$ $\mu \mathrm{m}$, $t=1.7$ $\mu \mathrm{m}$, and $L=4$ $\mu \mathrm{m}$ measured at $T=1.5$ K. Panel (a) shows the modulation of the magnetoconductance curves varying the sheet conductance ${\sigma }_{2\mathrm{D}}^0$ from 0.15 mS (top curve) to 1.3 mS (bottom curve). Panel (b) shows the inelastic (${\tau }_i$) and spin-orbit (${\tau }_{\mathrm{so}}$) scattering times extracted from the curves.

Figure 4

Magnetoconductance curves as a function of the temperature for selected doping levels, identified by the sheet conductance ${\sigma }_{2\mathrm{D}}^0$. Panel (a) shows curves acquired in a flow cryostat in the temperature range 1.5 K to 10 K, panel (b) shows curves acquired in a Heliox system with a base temperature of 0.3 K, and panel (c) shows curves acquired in a dilution cryostat keeping the temperature at 50 mK and changing the carrier concentration. Note the different field scales. Data in panels (a) and (b) refer to a 0.75 $\mu \mathrm{m}$ wide channel with $t=1.7$ $\mu \mathrm{m}$ and $L=4.5$ $\mu \mathrm{m}$, whereas data in panel (c) refer to a 0.5 $\mu \mathrm{m}$ wide channel with $t=1.7$ $\mu \mathrm{m}$ and $L=4.5$ $\mu \mathrm{m}$. The black solid lines in panels (a) and (b) are the fits performed using Eq. (1).

Figure 5

Evolution of the inelastic [${\tau }_i$ (blue triangles)], spin-orbit [${\tau }_{\mathrm{so}}$ (purple dots)] and elastic [${\tau }_{el}$ (yellow squares)] scattering times for the 0.75 $\mu \mathrm{m}$ wide channel as a function of the temperature for selected values of the sheet conductance. In the pink region spin-orbit interaction is the main scattering mechanism with ${\tau }_i>{\tau }_{\mathrm{so}}$. The crossing point between ${\tau }_i$ and ${\tau }_{\mathrm{so}}$ identifies the crossover temperature $T_{\mathrm{cross}}$ between WL and WAL.

Figure 6

Phase diagram showing the temperature and doping ranges for which LAO/STO magnetotransport is characterized by WAL or WL. The points represent the temperatures $T_{\mathrm{cross}}$ for which ${\tau }_i={\tau }_{\mathrm{so}}$. The different colors of the data points refer to side-gate field effect devices of different size: $w=1.3$ $\mu \mathrm{m}$, $t=1.7$ $\mu \mathrm{m}$, $L=8$ $\mu \mathrm{m}$ (green dots), $w=0.75$ $\mu \mathrm{m}$, $t=1.7$ $\mu \mathrm{m}$, $L=4.5$ $\mu \mathrm{m}$ (blue dots), $w=0.5$ $\mu \mathrm{m}$, $t=1.7$ $\mu \mathrm{m}$, $L=4.5$ $\mu \mathrm{m}$ (purple dot), $w=0.25$ $\mu \mathrm{m}$, $t=0.7$ $\mu \mathrm{m}$, $L=2.5$ $\mu \mathrm{m}$ (orange dot). The blue star is the crossing point extracted from the data of Fig. 3, in agreement with the results obtained in Refs. [9] and [10]. The brown dots refer to a sample with $w=500$ $\mu \mathrm{m}$, $L=1$ mm in the conventional back-gate configuration. The blue area at the bottom indicates the superconducting dome (right axis for the superconducting $T_c$). The inset is a zoom of the low-doping region. For ${\sigma }_{2\mathrm{D}}^0=0.05$ mS we obtained $T_{\mathrm{cross}}=50$ mK [see panel (c) of Fig. 4].