Phonons near charged defects
It is often assumed that vibrational contributions to the formation energy are negligible. But is this really true? We investigated this issue for a prototypical defect: the O +2 vacancy in MgO. Due to a significant red-shift of phonons near the vacancy, not only the zero-point vibrations are reduced by more than 0.1 eV (which normally would be expected only for hydrogen), but also the vibrational entropy increases. Thus, both the zero-point vibrations as well as the finite temperature effect act in favor of defect formation.
Motivation
The first approximation to understand defect formation with ab initio calculations is to neglect temperature effects and consider only the T=0K equilibrium geometry. In view of the challenges from matching computational ressources, defect models, and the required level of theory, it was often assumed that vibrational contributions to the energy are negligible. But is this really true? In view of recent advances in increasing the accuracy if the T=0K treatment of defects and the fact that charged defects may interact with a large number of nearby ions, we decided to investigate this issue for a prototypical defect: the O +2 vacancy in MgO.
Method
We developed an electro-elastic (or Coulomb-Hook) model to describe the phonons, and parameterized it from density-functional theory (DFT) in the local-density approximation (LDA). After generating a parameter set for the bulk material, we recomputed Born effective charges and spring constants for atoms near the vacancy from a 32 atom cell. It turned out that the most important changes occured for the first two shells of atoms. We then transferred the parameters to larger supercells to reduce defect-defect interactions.
It turned out that increasing the supercell for the actual phonon calculations is very important because small supercell show significant artifacts. The phonon DOS, or more precisely: its change when introducing the vacancy in the perfect bulk, stabilizes only for >200 atoms. Calculating the Hesse matrix for such a large system within DFT would normally be a formidable task. Using our model, it is achieved within minutes.
Results
The oxygen vacancy induces a complex change in the vibrational DOS which energetically overlaps with the bulk phonons. It will therefore be very challenging to detect in experiment. The shape of the defect DOS is significantly more complex than the bulk, and can also not be understood from the oxygen-only contribution to the bulk vibrations. Instead, a detailed analysis reveals that a major feature consists in a significant red-shift of the main peaks in the DOS, augmented by a few local modes towards the upper edge of the bulk DOS. The impact on the DOS exceeds the removed degrees of freedom by almost a factor five. In other words, the most remarkable changes indeed arise from the atoms nearby the vacancy.
The red-shift can be explained as follows. In the bulk, the lower feature of the DOS is dominated by Mg vibrations, while the upper feature arises from O vibrations. The removal of the oxide ion weakens the oxygen-oxygen interactions, while the reduction in the effective charge of the Mg ions next to the vacancy causes the Mg-Mg interactions to weaken. Therefore, both features are red-shifted, which appears as a positive/negative double feature in the DOS change.
The red-shift has remarkable consequences for the thermal properties. Not only the zero-point vibrations are reduced by more than 0.1 eV (which normally would be expected only for hydrogen), but also the vibrational entropy increases. The latter is counter-intuitive, since upon removing degrees of freedom, one would normally expect a reduction in the vibrational entropy. Thus, both the zero-point vibrations as well as the finite temperature effect act in favor of defect formation.