Metadynamics simulations of strontium-vacancy diffusion in ${\mathrm{SrTiO}}_3$

The sluggish diffusion of strontium vacancies in the perovskite-oxide ${\mathrm{SrTiO}}_3$ was studied by means of classical metadynamics simulations. Two different mechanisms were examined: the migration of an isolated strontium vacancy and the migration of a strontium vacancy in a defect associate with an oxygen vacancy. Combining activation Gibbs energies of migration with the appropriate temperature-dependent pre-exponential terms, we obtained diffusivities of strontium vacancies by the two mechanisms for the temperature range $1000\le T/K\le 2000$. Comparisons with experimental data (where available) yield excellent agreement, both in terms of the effective activation enthalpy of migration as well as the absolute rate of diffusion. In this way we demonstrate the ability of classical metadynamics simulations to predict diffusion coefficients quantitatively for extremely slow-moving defects in oxides, and we highlight the complexity of cation diffusion in perovskite materials.

Figure 1

Migration pathways (dashed orange lines) for the cases of strontium-ion migration (orange) into an isolated strontium vacancy (a) and into a strontium vacancy with an adjacent oxygen vacancy (b), as obtained by NEB calculations. Oxygen ions shown in grey, titanium ions in blue. Both paths are confined to the $xy$ plane of the Cartesian coordinate system, which allows us to express $\xi$ in terms of $x$ and $y$.

Figure 2

Convergence of the Gibbs energy surface during metadynamics simulations of the jump of a strontium cation into an isloated strontium vacancy with two parameter sets: A, $\omega =0.12\mathrm{eV}$ and B, $\omega =0.01\mathrm{eV}$ ($\sigma =0.2\mathring{\mathrm{A}}$ and $\delta t=100\mathrm{fs}$ in both cases).

Figure 3

Averaged Gibbs energy surfaces $G\left(\xi \right)$ along the $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ collective coordinates for the ${\mathrm{v}}_{\mathrm{Sr}}^{″}$ migration without (a), (b) and with adjacent ${\mathrm{v}}_{\mathrm{O}}^{••}$ (c), (d) as obtained by metadynamics simulations at $T=1100\mathrm{K}$. The starting point of the migration process, ${\xi }^i$, is marked with an orange dot and the top of the migration barrier, ${\xi }^{*}$, with a blue dot. The 2D mappings of the surface (b), (d) show that the adjacent ${\mathrm{v}}_{\mathrm{O}}^{••}$ causes a bending in the migration pathway of the Sr ion.

Figure 4

Temperature dependence of (a) the activation Gibbs energy of migration, $\mathrm{\Delta }{G}_{\mathrm{mig},\mathrm{v}}^{‡}$, as obtained by metadynamics; (b) the probability densities at the initial system configuration ${\xi }^i$, $P\left({\xi }^i\right)$, as calculated by straightforward MD; and (c) the conditional ensemble average of the collective coordinate velocity $\dot{\xi }$, $\langle |\dot{\xi }{|\rangle }_{\xi \left(0\right)={\xi }^{*}}$, as computed by constrained MD simulations.

Figure 5

Calculated strontium-vacancy diffusion coefficients $D_{\mathrm{v}}$ in ${\mathrm{SrTiO}}_3$ with and without an adjacent oxygen vacancy compared with experimental results by Meyer et al. [29] for $0.2\mathrm{at}\%$ Nd-doped (A) and by Gömann et al. [30] for $1.00\mathrm{at}\%$ La-doped (B) and $0.02\mathrm{at}\%$ La-doped (C) compositions.