Fig. 4: Na⁺ kinetics in P2/P3 Na_2/3Ni_1/3Mn_2/3O₂ layered oxides.

a, b Crystal environments of NaO₆ prism in both P2 and P3 phases, where the local coordination of the Na_f (2b, faces-sharing) and Na_e (2d, edge-sharing) sites in P2-phase and the Na (3a, face and edge-sharing) site in P3-phase are shown. c Calculated formation energies of Na_e and Na_f sites in a simple P2-NaCoO₂ model. d Energy differences of the in-plane Na⁺ ordering in P2-Na_2/3Ni_1/3Mn_2/3O₂ structure, where the Mn⁴⁺ and Ni²⁺ ions described by a (\(\surd 3\times \surd 3\))-R30° supercell in transition metal layer. e MSDs for Na-ions in P2- Na_2/3Ni_1/3Mn_2/3O₂ with Na⁺ LZZ ordering and disordering at 800 K, including the Disorder 1, [occ. (Na_f) = 1/6] and Disorder 2, [occ. (Na_f) = 1/12]. f Total MSDs for Na-ions in P2-Na_2/3Ni_1/3Mn_2/3O₂ with different occupations of Na⁺ at 800 K. Isosurface of the probability density distribution P(r) of Na⁺ in P2-Na_2/3Ni_1/3Mn_2/3O₂ with g honeycomb, [occ. (Na_f) = 0] h chain, [occ. (Na_f) = 1/2] and i LZZ [occ. (Na_f) = 1/6] Na ion orderings within the Na slabs at 800 K, and the isosurface level is set to 0.001, where the green ball represents the Na_f site, and the yellow one represents the Na_e site. j Migration energy barriers for diffusion trajectories in layered P2-Na_2/3Ni_1/3Mn_2/3O₂ with chain and k LZZ orderings. l Illustration of Path 1 in layered P2-Na_2/3Ni_1/3Mn_2/3O₂ with LZZ ordering in k.