Electronic structure and polaronic charge distributions of Fe vacancy clusters in ${\mathrm{Fe}}_{1-x}\mathrm{O}$

We perform a detailed study of the electronic structure of ${\mathrm{Fe}}_{1-x}\mathrm{O}$ at moderate values of $x$. Our results evidence that the Fe vacancies introduce significant local modifications of the structural, electronic, and magnetic features, which serve to explain the origin of the measured dependencies of the physical properties on $x$. The final properties are determined by a complex interplay of the charge demand from O, the magnetic interactions, and the charge order at the Fe sublattice. Furthermore, polaronic distributions of charge resembling those at magnetite, ${\mathrm{Fe}}_3{\mathrm{O}}_4$, emerge for the most stable defect structures. This defines a unique scenario to understand the nature of the short-range correlations in ${\mathrm{Fe}}_3{\mathrm{O}}_4$, and unveils their intimate connection to the long-range charge order developed below the Verwey transition temperature.

Figure 1

(a) Cubic unit cell of 64 atoms used to model FeO. Fe sites with opposite spin orientation are coloured in blue and green. (b) Same for a supercell with a 4:1 cluster. (c) Top view of three (111) planes around an Fe vacancy in ${\mathrm{Fe}}_{0.97}\mathrm{O}$. The central thick hexagon joins the Fe atoms surrounding the vacancy site in the blue layer.

Figure 2

Spin-resolved DOS of stoichiometric FeO with positive (negative) values corresponding to majority (minority) spins. The black curve corresponds to the total DOS of the supercell, the blue one to the Fe contribution.

Figure 3

Histograms showing the distribution of values of d(Fe-Fe) at the ${\mathrm{Fe}}_B$ sublattice and its decomposition in $d_{\parallel }$ and $d_{\perp }$ for the structures in Table 1. The vertical scale of the middle and bottom panels has been reduced by $\sim 30\%$ with respect to the top panel.

Figure 4

Energy cost of creating a defect at the different structures considered in Table 1, as a function of ${\mu }_O$.

Figure 5

(a) Total DOS of ${\mathrm{Fe}}_{0.97}\mathrm{O}$ (black curve) with positive (negative) values corresponding to majority (minority) spins. The blue and purple curves are the projections on the $\mathrm{Fe}{{}_B}^{2+}$ and $\mathrm{Fe}{{}_B}^{3+}$ sites, respectively. (b) Charge density around an $\mathrm{Fe}{{}_B}^{3+}$ site, showing the accumulation of charge at certain $\mathrm{Fe}{{}_B}^{3+}-\mathrm{Fe}{{}_B}^{2+}$ connecting lines. (c) DOS around the Fermi level of the $\mathrm{Fe}{{}_B}^{2+}$ pointed by an arrow in (b), showing the states directed along the CL that overlap with those of $\mathrm{Fe}{{}_B}^{3+}$. The vertical scale is 10 times that in (a).

Figure 6

Top view of two adjacent Fe(111) planes of ${\mathrm{Fe}}_{0.906}\mathrm{O}$ with three isolated $V_{\mathrm{Fe}}$, under configurations C1 (left) and C2 (right). The lines join the six in-plane n.n. of each $V_{\mathrm{Fe}}$. The legend of Fig. 1 has been used, and the O sublattice has been omitted for clarity.

Figure 7

Same as Fig. 5 for the supercells modeling ${\mathrm{Fe}}_{0.906}\mathrm{O}$ with three isolated $V_{\mathrm{Fe}}$. Top (bottom) panel corresponds to the C1 (C2) configuration.

Figure 8

Side view along $c$ (left) and top view (right) of ${\mathrm{Fe}}_{0.906}\mathrm{O}$ around a 4:1 cluster, following the legend in Fig. 1. On the right, only two adjacent (111) Fe layers are shown, and the ${\mathrm{Fe}}_A$ atom lies between them. The thick lines join the in-plane n.n. to the vacancy site.

Figure 9

Same as Fig. 5 for the supercell modeling 4:1 clusters in ${\mathrm{Fe}}_{0.906}\mathrm{O}$. The red curve is the projection on the tetrahedral $\mathrm{Fe}{{}_A}^{3+}$ atom.

Figure 10

Two-dimensional slice of the charge density at the (111) plane containing the $V_{\mathrm{Fe}}$ in ${\mathrm{Fe}}_{0.97}\mathrm{O}$ (right), showing on the left the reference atomic positions. Representative interatomic distances (in $\text{Å})$ between ${\mathrm{Fe}}^{2+}-{\mathrm{Fe}}^{3+}$ pairs are provided, and the corresponding CLs indicated by the dashed lines on the CD plot.

Figure 11

(a) Sketch of a $P2/c$ unit cell of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ identifying the trimerons. The legend of Fig. 1 has been used. (b) Top view of a plane containing two trimerons. (c) Charge density at the plane in (b). The dashed lines follow the rows defined by the ${\mathrm{Fe}}_B$ atoms.

Figure 12

Same as Fig. 10 for case C1 of ${\mathrm{Fe}}_{0.906}\mathrm{O}$, at the (111) Fe plane where the shortest d(Fe-Fe) can be found.

Figure 13

CD around a 4:1 defect cluster at the two different types of (111) planes in ${\mathrm{Fe}}_{0.906}\mathrm{O}$ (top). The reference atomic positions are indicated by dashed lines on top of the CD and at the lower panel, that also shows the projection of the ${\mathrm{Fe}}_A$ position. Representative values of the interatomic ${\mathrm{Fe}}^{2+}-{\mathrm{Fe}}^{3+}$ distances are provided (in $\text{Å})$.