Vibrational entropies of oxygen vacancy formation in metal oxides

Oxygen vacancies play a key role in many energy-related applications, and the investigation of the thermodynamic forces driving their formation across different materials can be tackled through the use of ab initio methods. In this context, however, computational studies targeting finite-temperature properties such as entropy remain scarce. Wider-ranging studies are of interest to deepen the understanding of the role of the entropy of oxygen vacancy formation and its different components in processes like the ones involved in solar thermochemical hydrogen (STCH) production, and to investigate the existence of common trends and differences between materials. In this work, we use density functional theory and harmonic phonon calculations to compute the vibrational entropy of oxygen vacancy formation ($\mathrm{\ensuremath{\Delta}}{S}_{\text{vib}}$) for 10 different metal oxide compounds. The computation is carried out by taking the difference between the vibrational entropy of a vacancy-containing and a pristine structure, while the entropic contribution from the final gaseous state of the oxygen lost from the material is accounted for separately with a gas entropy term. We first examine the temperature dependence of $\mathrm{\ensuremath{\Delta}}{S}_{\text{vib}}$ and highlight the presence of an initial peak around room temperature, followed by a steady decrease resulting from the presence of an additional O atom (the one becoming vacant) in the defect-free structure. We then inspect the atomic contributions to $\mathrm{\ensuremath{\Delta}}{S}_{\text{vib}}$, and highlight similarities and differences between compounds. Finally, we consider other significant sources of entropy, and find $\mathrm{\ensuremath{\Delta}}{S}_{\text{vib}}$ to provide a smaller, yet non-negligible, contribution to the total entropy of vacancy formation. We also compare the temperature dependence of $\mathrm{\ensuremath{\Delta}}{S}_{\text{vib}}$ to that of the gas entropy, and show that the two largely counterbalance each other at high temperatures.