Fig. 2. Electron-polaron migration in hybrid-functional DFT.

(A) The isodensity of an electron polaron in bulk hematite Fe₂O₃. At the initial state, the excess electron is mostly localized on Fe atoms marked 0 and 1. The polaron can migrate within the plane of Fe atoms with the same spin (in-plane hop) to atoms 1,2 or across the plane with the opposite spin (inter-plane hop) to atoms 3,4. O atoms are shown in pink, and Fe in brown and gray to distinguish between opposite orientations of the local magnetic moments. (B) Spatial distribution of a delocalized electron; the charge in the lowest (partially) occupied band at the bottom of the conduction band is shown. Panels (C) to (E) show the energy barriers as a function of the progressing lattice distortion for in-plane (C) and interplane (D) polaron hopping, as well as for the delocalization process (E). Panels (F) to (H) show the evolution of the spin-resolved density of states during the diffusion processes in the top panels.