Orbital ordering in the ferromagnetic insulator ${\mathrm{Cs}}_2\mathrm{Ag}{\mathrm{F}}_4$ from first principles

We found, using density-functional theory calculations within the generalized gradient approximation, that ${\mathrm{Cs}}_2\mathrm{Ag}{\mathrm{F}}_4$ is stabilized in the insulating orthorhombic phase rather than in the metallic tetragonal phase. The lattice distortion present in the orthorhombic phase corresponds to the $x^2-z^2∕y^2-z^2$ hole-orbital ordering of the ${\mathrm{Ag}}^{2+}4d^9$ ions, and this orbital ordering leads to the observed ferromagnetism, as confirmed by the present total-energy calculations. This picture holds in the presence of moderate $4d$-electron correlation. The results are compared with the picture of ferromagnetism based on the metallic tetragonal phase.

Figure 1

(Color online) Density of states (DOS) of ${\mathrm{Cs}}_2\mathrm{Ag}{\mathrm{F}}_4$ in the FM orthorhombic phase from GGA. This insulating solution has a small band gap of $0.2\mathrm{eV}$. The Fermi level is set at zero energy. Solid red (dashed blue) lines depict the spin-up (-down) states. The panels show, from top to bottom, the total DOS per f.u. and orbital-resolved DOS. The state around $-7\mathrm{eV}$ comes from the $\mathrm{Cs}5p$ orbital. For the $\mathrm{Ag}4d$-resolved DOS, the local coordinate system is chosen in such a way that the $y$ $\left(x\right)$-axis is along the in-plane longer (shorter) Ag-F bond. The ${\mathrm{Ag}}^{2+}$ $x^2-z^2$ hole state and the corresponding $y^2-z^2$ hole state of the nearest-neighbor ${\mathrm{Ag}}^{2+}$ ions form the in-plane orbital ordering (see also Fig. 2).

Figure 2

(Color online) Contour plot $(0.1–0.8e∕{\mathrm{\AA }}^3)$ of the spin density in the $xz$ (upper panel) and $yz$ (lower panel) planes through the $\mathrm{Ag}{\mathrm{F}}_6$ octahedra of ${\mathrm{Cs}}_2\mathrm{Ag}{\mathrm{F}}_4$ in the FM insulating orthorhombic phase obtained by GGA. It evidences the $x^2-z^2∕y^2-z^2$ hole-orbital ordering of the ${\mathrm{Ag}}^{2+}$ ions in the $xy$ basal plane. See also Fig. 1. Note also a relatively strong spin polarization of the F atoms (see more in the text).

Figure 3

(Color online) Orbital-resolved DOS of ${\mathrm{Cs}}_2\mathrm{Ag}{\mathrm{F}}_4$ in the FM tetragonal phase from GGA. It is nearly half-metallic with the wide in-plane $x^2-y^2$ band and the small $e_g$-level splitting.

Figure 4

(Color online) DOS of ${\mathrm{Cs}}_2\mathrm{Ag}{\mathrm{F}}_4$ in the FM orthorhombic phase from $\mathrm{GGA}+U$ with $U=3\mathrm{eV}$. The insulating gap is increased up to $1.0\mathrm{eV}$, compared with GGA (see Fig. 1).