Dynamics of the spatial separation of electrons and mobile oxygen vacancies in oxide heterostructures

In the search for an oxide-based 2D electron system with a large concentration of highly mobile electrons, a promising strategy is to introduce electrons through donor doping while spatially separating electrons and donors to prevent scattering. In $\mathrm{SrTi}{\mathrm{O}}_3$, this can be achieved by tailoring the oxygen vacancy profile through reduction, e.g., by creating an interface with an oxygen scavenging layer. Through reduction, oxygen atoms are removed close to the interface, leaving behind oxygen vacancies in the $\mathrm{SrTi}{\mathrm{O}}_3$ lattice and mobile electrons in the $\mathrm{SrTi}{\mathrm{O}}_3$ conduction band. The commonly assumed picture is that the oxygen vacancies then remain confined close to the interface while the electrons leak a few nanometers into the bulk, resulting in an electron-defect separation and a highly mobile, oxide-based 2D electron system. So far it has remained unclear how the confinement and electron-defect separation develop over time. Here, we present transient finite element simulations that consider three driving forces acting on the oxygen vacancy distribution: diffusion due to the concentration gradient, drift due to the intrinsic electric field, and an oxygen vacancy trapping energy that holds oxygen vacancies at the interface. Our simulations show that at room temperature, three distinct regions are formed in $\mathrm{SrTi}{\mathrm{O}}_3$ within days: (1) Oxygen vacancies are partially held at the interface due to the oxygen vacancy trapping energy. (2) The accompanying positive space charge causes an oxygen vacancy depletion layer with large electron concentration and high mobility just below the interface. This electron-defect separation, indeed, leads to a highly conductive region. (3) While we are able to describe measured conductivity data with an oxygen vacancy trapping energy of $-0.2$ eV, this value does not prevent oxygen vacancy diffusion into the bulk: A diffusion front progresses into the bulk and leads to significant conductivity arising over the first micrometer within a couple of months. An enhanced oxygen vacancy trapping energy of $-0.5$ eV or below would suppress this loss of confinement, leading to a static and pronounced electron-defect separation. Consequently, our results highlight the importance of oxygen vacancy redistribution and suggest the trapping energy of oxygen vacancies at the interface as an important design parameter for oxygen-vacancy-based 2D electron systems.

Figure 1

(a) The assumed geometry is 3.5 mm of $\mathrm{SrTi}{\mathrm{O}}_3$ capped with an oxygen scavenging layer, e.g., $\gamma \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$. (b) The assumed profiles for the initial oxygen vacancy distribution (red) and the background of acceptor-type impurities (purple). The arrows indicate the effect of the three driving forces acting on the oxygen vacancy profile: Due to the oxygen vacancy trapping energy (turquoise arrow), oxygen vacancies are pulled to the interface, while diffusion (red arrow) and electrostatic potential (green arrow) lead to a redistribution into the bulk. (c) The initial oxygen vacancy distribution next to the resulting initial electron profile (blue) at 300 K. The separation of electrons and donors becomes apparent. (d) The profile of the standard chemical potential for oxygen vacancies relative to the bulk. At the interface, this potential is lowered due to the oxygen vacancy trapping energy. (e) The electrostatic potential profile caused by the spatial separation of electrons and donors. The resulting electric field pulls oxygen vacancies into the bulk.

Figure 2

Simulated redistribution of oxygen vacancies at room temperature for an oxygen vacancy trapping energy of $\mathrm{\Delta }{g}_v^0=-0.2$ eV. (a) Development of oxygen vacancy concentration $v$ (red), electron concentration $n$ (blue), and the electrostatic potential $\phi$ (green) over 9 months of storage at room temperature. The initial profile as well as the profiles after 14 days and 9 months are colored in a light, medium, and darker shade, respectively. Profiles of intermediate times are shown in gray. The assumed concentration of acceptor-type impurities is included for reference (purple). (b) Dotted lines: Measured temperature dependence of sheet resistance $R_S$, sheet carrier density $n_S$, and mobility $\mu$ before (green) and after (red) several months of storage at room temperature [47]. Solid lines: Comparable data calculated from the simulated concentration profiles after 14 days and 9 months. The change of the low-temperature mobility with time is shown in Fig. 3.

Figure 3

Effect of oxygen vacancy redistribution on the low-temperature mobility, assuming an oxygen vacancy trapping energy of $\mathrm{\Delta }{g}_v^0=-0.2$ eV. (a) Development of the mobility at 2 K calculated from the simulated concentration profiles (circles) next to measured data (diamonds) from Ref. [47]. The measurements were taken 13 days (green) and 8 months (red) after deposition. The data points from simulations for 14 days and 9 months that are compared to the measurements are colored accordingly. (b) Top: Low-temperature (2 K) concentration profiles of oxygen vacancies $v$ and electrons $n$ after 14 days (lighter color) and 9 months (darker color) of storage at room temperature. These concentrations determine the local mobility $\mu$ (second plot from top) and the local conductivity $\sigma$ (third plot from top). Bottom: The integral of the local conductivity (conductance up to $x$) is plotted as a function of $x$.

Figure 4

(a) Influence of the oxygen vacancy trapping energy $\mathrm{\Delta }{g}_v^0$ on the development of the low-temperature mobility. (b) For $\mathrm{\Delta }{g}_v^0=-0.5\mathrm{eV},T=300\mathrm{K}$: Development of oxygen vacancy concentration $v$ (red), electron concentration $n$ (blue), and the electrostatic potential $\phi$ (green) over 9 months of storage at room temperature. The assumed concentration of acceptor-type impurities is included for reference (purple). (c) For $\mathrm{\Delta }{g}_v^0=-0.5\mathrm{eV},T=2\mathrm{K}$: Low-temperature concentration profiles of oxygen vacancies $v$ and electrons $n$ after 14 days (lighter color) and 9 months (darker color) of storage at room temperature. These concentrations (top) determine the local mobility $\mu$ (second from top) and local conductivity $\sigma$ (third from top). Bottom: The integral of the local conductivity (conductance up to $x$) is plotted as a function of $x$.