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 $\ensuremath{-}0.2$ 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 $\ensuremath{-}0.5$ 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.