The Diffusion Kinetics of Ba Cations in Perovskite BaTiO$_3$: A Combined Tracer Diffusion and Metadynamics Study

Tracer diffusion experiments and metadynamics (MtD) simulations were used to study the diffusion of Ba cations in the cubic phase of the perovskite oxide BaTiO$_3$. $^{130}$BaTiO$_3$ thin films were used as diffusion sources to introduce barium tracer diffusion profiles into single-crystal samples at temperatures $1348 \leq T/\mathrm{K} \leq 1498$. The $^{130}$Ba profiles were determined by time-of-flight secondary ion mass spectrometry, and then analysed to yield Ba tracer diffusion coefficients ($D_\mathrm{Ba}^\ast$). MtD simulations were performed in order to obtain barium-vacancy diffusion coefficients ($D_\mathrm{v_{Ba}}$) for selected vacancy mechanisms as a function of temperature. $D_\mathrm{v_{Ba}}$ is predicted to be increased significantly by an adjacent oxygen vacancy, and even more, by an adjacent titanium vacancy. From the combined consideration of $D_\mathrm{Ba}^\ast$ and $D_\mathrm{v_{Ba}}$, we conclude that Ba diffusion in these samples occurred most probably by the migration of defect associates, and not by the migration of isolated barium vacancies. More generally, our results draw attention to the dangers of relying solely on activation enthalpies to interpret diffusion data.