Permutation-invariant collective variable to track and drive vacancy dynamics in simulations of solids

Vacancy dynamics in oxides are vital for understanding redox reactions and resulting memristive effects or catalytic activity. We present a method to track and drive vacancies which we apply to metadynamics simulation of oxygen vacancies (V${}_{\mathrm{O}}{}^{2+}$) in rutile, demonstrating its effectiveness. Using the density-functional based tight-binding method, it is possible to explore the free energy hyperplane of oxygen vacancies in TiO${}_2$. We show that the migration of V${}_{\mathrm{O}}{}^{2+}$ in TiO${}_2$ is governed by the jump with the highest degree of topological interconnection. Free energy profiles are consistent with minimum energy paths.

Figure 1

(a) 130 000 step free energy profile of V${}_{\mathrm{O}}{}^{2+}$ along (001) in $2\times 2\times 3$ supercell. Wall potentials constrain the motion of the vacancy at 1.7 and 5.8 Å. (b) Vacancy position shown every 100 fs as (dark) blue spheres, viewed approximately along the (100) axis; Ti and O atoms of the minimum energy structure of the TiO${}_2$ cell are shown for orientation. (c) Cutout of the the FEP in the $3\times 3\times 6$ supercell after 75 000 steps.

Figure 2

Possible transitions for a V${}_{\mathrm{O}}$ defect in rutile. Initial vacancy position is marked in black.