${}^{18}$O/${}^{16}$O exchange and subsequent time-of-flight secondary ion mass spectrometry (ToF-SIMS) analysis was employed to investigate the transport of oxygen, and thus the behavior of oxygen vacancies, in [nominally undoped, (100) oriented] single-crystal SrTiO${}_3$ substrates. Isotope exchange anneals were performed as a function of temperature, 948 $<$ $T$/K $<$ 1123, at an oxygen activity $a$O${}_2$ $=$ 0.50 and as a function of oxygen activity, 0.01 $<$ $a$O${}_2$ $<$ 0.70, at $T$ $=$ 1073 K. All isotope profiles show the same characteristic form: an initial drop over tens of nanometers close to the surface, which is attributed to an equilibrium space-charge layer depleted of oxygen vacancies, followed by a profile extending several microns into the solid, which is attributed to diffusion in a homogeneous bulk phase. The entire isotope profile can be described quantitatively by a numerical solution to the diffusion equation with a position-dependent diffusion coefficient; the description yields the tracer diffusion coefficient in the bulk $D^{*}$(∞), the surface exchange coefficient $k_{\mathrm{s}}^{*}$, and the space-charge potential ${\Phi }_0$. All $D^{*}$(∞) data are consistent with nominally undoped SrTiO${}_3$ substrates being weakly acceptor doped; the activation enthalpy for the migration of oxygen vacancies in bulk SrTiO${}_3$ is found to be $\Delta H_{\mathrm{mig},\mathrm{V}}$ ≈ 0.6 eV. The surface termination of the SrTiO${}_3$ substrates was seen to affect significantly the surface exchange coefficient $k_{\mathrm{s}}^{*}$. Values of ${\Phi }_0$ obtained as a function of $T$ and $a$O${}_2$ are approximately 0.5 V, indicating strong depletion of oxygen vacancies within the equilibrium surface space-charge layers. Thermodynamic modeling indicates that space-charge formation at the TiO${}_2$-terminated (100) surface is driven by the Gibbs formation energy of oxygen vacancies at the interface being lower than in the bulk.

• Received 16 March 2012

DOI:https://doi.org/10.1103/PhysRevB.85.174109