2.8 Bader analysis
3.1.4 Simulated defect migration barriers
3.1.4.1 Interatomic potentials
DFT, semi-empirical and interatomic potential calculations have been used to calculate the migration barrier of defects in MgO, using either the constrained minimisation or nudged elastic band (NEB) techniques. Busker et al. used Buckingham potentials to study the doubly-charged defects of MgO. The potentials were fitted to the lattice parameters, elastic and dielectric constant, for fully charged and partially charged (±1.7e) systems with the shell model.[84] The migration barrier calculated for the doubly charged oxygen (F2+) vacancy is 2.12 eV using the fully charged model and 1.56 eV for the partial charge model.[84] The fully charged barrier is in better agreement with the experimentally observed barrier for this vacancy, while the partially charged model differs by 35.5% from the experimental value. There is a similar difference between the barriers calculated for the Mg vacancy (V2−), with a barrier of 2.08 eV for the fully charged potentials, and 1.53 eV for the partially charged potentials.[84]The barriers of the vacancies predicted by the different potentials are similar because of the migration pathways the vacancies take. The interstitial migration barriers were also calculated using these different potentials. The difference in the predicted barrier heights was not so significant. The O2−interstitial migration barriers were calculated to be 0.58 eV and 0.47 eV, while the Mg2+ interstitial migration barrier was 0.64 eV and 0.46 eV using the fully and partially charged models, respectively. In 2005 Uberuaga et al. used Buckingham potentials to study the same defects using the interatomic potentials derived by Lewis and Catlow,[85] which is a fully charged model. The migration barrier was calculated using the NEB method and resulted in a barrier height of 2.00 eV and 2.12 eV for the F2+ and V2− vacancies, respectively.[86] These calculated barriers are in good agreement with those calculated by Busker et al. when using the fully charged model and the experimental results for the F2+vacancy. Although the vacancy barrier heights are in agreement
with these two different potentials, there is a difference when considering the migration of the interstitials. Uberuaga et al. has calculated migration barriers of 0.40 eV and 0.32 eV for the O2−and Mg2+ interstitials, respectively.[86]The lower barriers calculated by Uberuaga et al. could be due to the NEB method implemented to calculate the pathway. As the NEB method optimises the pathway, allowing full relaxation of the saddle point, unlike the method used by Busker et al., it may find a lower energy saddle point.
3.1.4.2 Semi-empirical method
Kotomin and Popov used semi-empirical methods to investigate the diffusion of the oxygen vacancies in different charge states in a 223 atom cluster, using the constrained minimisation technique by fixing the oxygen between the two vacancies along the (110) axis.[87] The semi-empirical method is based on the formulation of Hartree-Fock (HF) and the shell model pair potential. The migration barrier calculated for the F2+vacancy is 2.50 eV, which is higher than that predicted by the interatomic potentials but is in better agreement with the range observed from experiments. This computational technique allows different charge states to be studied, and therefore the migration barriers of the oxygen vacancies with one (F+) and two (F0) electrons localised in the vacancy. The migration barrier of the F+vacancy was calculated to be 2.72 eV. Therefore, the localised electron results in an increase in the migration barrier of 0.22 eV. Localisation of another electron results in a migration
barrier of 3.13 eV, with the second electron resulting in a 0.41 eV increase in the barrier height.[87] There is no explanation as to why the introduction of a second electron causes an increase in the barrier height which is almost double that of the increase caused by one electron. There has been no experimental investigation of the diffusion of the oxygen vacancy with trapped electrons, so there is no way of assessing the accuracy of these semi-empirical calculations. The calculated migration barrier for the V2− vacancy was 2.43 eV,[87] which is also higher than predicted by the empirical potentials. This technique was also used to calculate the migration barriers of the interstitial defects in MgO. The lowest energy pathways for both doubly-charged interstitial species was the ion exchange mechanism, which involves the interstitial moving in the <111> plane and pushing the lattice ion into the new interstitial position while the interstitial takes the lattice site. The migration barrier for the Mg2+interstitial is 0.43 eV.[87] This migration barrier is in agreement with the results obtained by Uberuaga[86] and Busker et al.[84] through potentials, which shows that the empirical potentials can give a good description of defects in MgO. The O2− interstitial is also in agreement, with the migration
barrier calculated to be 0.54 eV.[87] The O22− interstitial diffusion barrier was also calculated by Kotomin
and Popov using two different methods, a 16 atom supercell using the HF method and a full-potential linear- muffin-tin-orbitals (FP LMTO) method based on the local density approximation (LDA). The calculated barrier is 1.45 eV for the LMTO method and 2.3 eV by the HF method.[87] The difference between the computational methods is accounted for by the difference in simulation cell size and the number of surrounding ions able to relax. However, both simulations found that the O22−interstitial has an interesting diffusion pathway, involving
the rotation of the dumb-bell and diffusion along the cube face. The <111> oxygen dumb-bell rotates to a <110> dumb-bell, with a barrier of 0.09 eV (HF) or 0.15 eV (LMTO). The dumb-bell then breaks and the interstitial then diffuses along the <110> face, with a migration barrier of 2.3 eV (HF) or 1.45 eV (LMTO). The interstitial oxygen then continues along the <110> face until it forms a <110> dumb-bell with another lattice oxygen, which then rotates back to the <111> dumb-bell with the same barriers as before.[87]
3.1.4.3 Density functional theory
The migration barrier of the doubly-charged defects has also been studied through DFT using the LDA. The migration barrier of the V2− vacancy is calculated to be 2.20 eV, which has a smaller migration barrier
than calculated for the F2+ vacancy of 2.31 eV.[88] The F2+migration barrier is smaller than that calculated by the semi-empirical method[87] but larger than those calculated by classical methods. [86, 84]The reason for the lower migration barrier calculated using DFT than the semi-empirical result could be due to the small simulation cell used by Kotomin and Popov. The doubly-charged interstitials have also been studied using two different sized simulation cells, with three different pathways investigated. The lowest energy pathway for the Mg2+interstitial was found to be the <111> ion exchange pathway, in support of the results obtained through classical results, with a barrier height of 0.83 eV (54 atom cell) and 0.71 eV (128 atom cell).[88] This results shows that the amount of relaxation in the surrounding ions has a significant effect on the migration barrier. The migration barrier of 0.71 eV is significantly higher than the barrier calculated using other computational methods. Whilst the O2− interstitial migration barrier is calculated to be 0.59 eV and 0.44 eV in the 54 and
128 cells, respectively, this agrees with the other computational methods. The barriers calculated by Kotomin and Popov and Uberuaga et al. found that the Mg2+interstitial should be more mobile than the O2−interstitial,
which disagrees with the results of Gilbert et al.[87, 86] The Gilbert et al. migration barrier is supported by the fully charged model used by Busker et al., whilst the partial charge model predicts the two interstitials to have equal migration barriers.[84]