Fig. 5.

Density functional theory modeling of atomic and electronic structures of bulk LSNF in ferromagnetic configuration. a The ground state 2 × 2 × 1 cell of La_0.5Sr_1.5NiO₄, with uniformly distributed La in each [001] AO layer. b Representations of the two B(B′)O₂ layers for each ground state La_0.5Sr_1.5Ni_1−xFe_xO₄, with x at 0, 0.25, 0.5, 0.75, and 1; for each x, the ground state Fe arrangement is characterized by equal numbers of Fe in the two B(B′)O₂ layers, with the Fe in each layer arranged to maximize the number of Fe–O–Ni bridges. c The corresponding spin polarized PDOS of e_g (Ni: gray shaded area and Fe: blue line) and 2p (O: red line) with respect to Fermi level ($E_{\mathrm{F}}^0$) for the B–O–B′ bridges; the PDOS are the average of existing B–O–B′ bridges; the adapted Fermi level ($E_{\mathrm{F}}^{\delta }$) to oxygen hyperstoichiometry is estimated via the rigid band model, with $E_{\mathrm{F}}^{\delta }=E_{\mathrm{F}}^0-2e^{-}\mathrm{\delta }∕\mathrm{DOS}\left(E_{\mathrm{F}}^0\right)$ (Eq. (1)), where $\mathrm{DOS}\left(E_{\mathrm{F}}^0\right)$ is the total density of states at $E_{\mathrm{F}}^0$ per formula unit of LSNF. d Top: Computed values of the O–p band center (top panel), with La₂NiO₄ calculated as reference. Middle: The magnitude of band overlap, determined by the integration of maximum PDOS between e_g(B) and e_g(B′) from −2 to 2 eV, normalized to e_g(Ni) in LSN. Bottom: The corresponding overlap center (centroid) of maximum PDOS between e_g(B) and e_g(B′). Additional details can be found in Supplementary Note 2