Ab initio calculation of the migration free energy of oxygen diffusion in pure and samarium-doped ceria

We have studied the free energy migration barriers Delta F-dagger for oxygen diffusion in pure ceria and Sm-doped ceria for the temperatures 300, 700, and 1000 K. We used the density functional theory in the generalized gradient approximation and an additional Hubbard U parameter for the Ce 4f electronic states. We compare the results for the free energy deduced from three different methods. First, a static harmonic approach is applied in which the temperature dependent vibrational contributions to energy and entropy are deduced from the phonon frequencies of supercells with a fixed volume. Second, a static quasiharmonic approach is used in which a part of the anharmonicity effect is introduced via an implicit dependence of the harmonic frequencies on the thermally expanding cell volume. Third, the free energy barriers are calculated using metadynamics and molecular dynamics in which anharmonicity effects are naturally taken into account. The three methods examined in this study lead to distinctly different results. According to the harmonic approximation, the migration free energy difference Delta F-double dagger increases with increasing temperature due to an increasing entropic contribution. According to the quasiharmonic approximation, the migration free energy is independent of temperature. Finally, molecular dynamics predicts a thermally induced increase in the migration free energy. We conclude that temperature dependent experimental lattice constants cancel out the increasing entropic contribution with increasing temperature in the static quasiharmonic approach. The full consideration of anharmonicity effects in the metadynamics method again leads to a temperature dependent migration free energy.