Origin of ferroelectricity in the multiferroic barium fluorides $\mathrm{Ba}M{\mathrm{F}}_4$: A first principles study

We present a first-principles study of the series of multiferroic barium fluorides with the composition $\mathrm{Ba}M{\mathrm{F}}_4$, where $M$ is Mn, Fe, Co, or Ni. We discuss trends in the structural, electronic, and magnetic properties, and we show that the ferroelectricity in these systems results from the freezing in of a single unstable polar phonon mode. In contrast to the case of the standard perovskite ferroelectrics, this structural distortion is not accompanied by charge transfer between cations and anions. Thus, the ferroelectric instability in the multiferroic barium fluorides arises solely due to size effects and the special geometrical constraints of the underlying crystal structure.

Figure 1

(Color online) (a) Projection of the $\mathrm{Ba}M{\mathrm{F}}_4$ structure along the $a$ axis. The $M$ cations are octahedrally surrounded by fluorine anions, which form puckered sheets perpendicular to the $b$ axis, separated by similar sheets of Ba cations. Adjacent sheets are shifted relative to each other by half a lattice constant along the $a$ direction. (b) Corresponding centrosymmetric prototype structure.

Figure 2

Total densities of states (gray shaded), partial fluorine $p$ (shaded with thin diagonal lines), and transition-metal $d$ (black shaded) states (in states/eV), calculated within the LSDA (left panel) and by using $U_{\mathrm{eff}}=4\mathrm{eV}$ (right panel) for all $\mathrm{Ba}M{\mathrm{F}}_4$ systems ($M=\mathrm{Mn}$, $\mathrm{Fe}$, $\mathrm{Co}$, $\mathrm{Ni}$ from top to bottom). Minority spin states are shown with negative sign. Zero energy corresponds to the Fermi level (metallic systems) or the highest occupied state (insulating systems).

Figure 3

(Color online) Magnetic structure of $\mathrm{Ba}M{\mathrm{F}}_4$, $M=\mathrm{Mn}$, $\mathrm{Fe}$, $\mathrm{Ni}$ (only magnetic ions are shown). For $M=\mathrm{Co}$ (phase $A$) the magnetic moments are oriented parallel to the $c$ axis, but the relative orientations of the moments are the same. Gray lines outline the conventional orthorhombic unit cell; the puckered rectangular grids are also indicated.

Figure 4

(Color online) Heisenberg nearest-neighbor exchange coupling constants $J_a$ (circles) and $J_c$ (triangles).

Figure 5

(Color online) Heisenberg coupling constants for ${\mathrm{BaNiF}}_4$ as a function of $U_{\mathrm{eff}}$ ($J_a$, circles; $J_c$, triangles). Open symbols correspond to calculations done with structural parameters obtained within the LSDA. Filled symbols correspond to fully relaxed calculations.

Figure 6

(Color online) Energy differences $\mathrm{\Delta }E$ per formula unit between the ferroelectric and the centrosymmetric prototype structures, calculated within the LSDA (circles) and by using $U_{\mathrm{eff}}=4\mathrm{eV}$ (triangles).