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(Color online) Ball representation of O sites in monoclinic corresponding to possible O vacancy sites. Threefold (yellow, light gray) and fourfold (blue, dark gray) coordinated sites are distinguished. Transitions to neighboring sites (thin dark lines) are indicated for sites and (subscripts are dropped in the figure). The highlighted one-dimensional chains correspond to lowest-energy pathways for O vacancy diffusion in the doubly positive charge state.
(Color online) Schematic energy profile along the lowest-energy pathway for long-range diffusion of the doubly positively charged oxygen vacancy in monoclinic . The sites are labeled according to the notation in Fig. 1.
Formation energy of the O vacancy (a) as a function of electron chemical potential in monoclinic and (b) along the coordinate perpendicular to the interface. In (a), is referenced with respect to the valence band of , and the Si band edges, and , are indicated. In (b), the neutral and two doubly positively charged states are considered, corresponds to the interface plane, and the lines are guides for the eye.
Activation energies for the migration of neutral and doubly positively charged oxygen vacancies between neighboring sites in monoclinic . The capital letters indicate O sites of the lattice and correspond to the labeling in Fig. 1. The subscripts III and IV indicate the coordination of the labeled site. For every transition, the hopping distance and the number of Hf atoms facing both the initial and final vacancy sites are given.
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