Magnetic-ion-induced displacive electric polarization in $\mathrm{Fe}{\mathrm{O}}_5$ bipyramidal units of $(\mathrm{Ba},\mathrm{Sr})\mathrm{F}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$ hexaferrites

Electric polarization in conventional paraelectric/ferroelectric oxides usually involves the displacement of nonmagnetic transition-metal ions with an empty $d$ shell. Here we unravel an unusual mechanism for electric polarization based on the displacement of magnetic ${\mathrm{Fe}}^{3+}$ $\left(3d^5\right)$ ions. Our simulations suggest that the competition between the long-range Coulomb interaction and short-range Pauli repulsion in a $\mathrm{Fe}{\mathrm{O}}_5$ bipyramidal unit with proper lattice parameters would favor an off-center displacement of ${\mathrm{Fe}}^{3+}$, which directly induces a local electric dipole. As a prototype example, we show that the electric dipole of a $\mathrm{Fe}{\mathrm{O}}_5$ bipyramid in $(\mathrm{Ba},\mathrm{Sr})\mathrm{F}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$ hexaferrites leads to a different family of magnetic quantum paraelectrics. The manipulation of this unique magnetic-ion-induced displacive electric polarization in a $\mathrm{Fe}{\mathrm{O}}_5$ bipyramid could open up a promising route to generating unconventional dielectrics, ferroelectrics, and multiferroics.

Figure 1

Crystal structures of $M$-, $Y$-, $W$-, and $Z$-type hexaferrites.

Figure 2

The c-axis dielectric permittivity ${\varepsilon }_c$ at 1 MHz as a function of temperature for (a) $\mathrm{BaF}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$, (c) $\mathrm{B}{\mathrm{a}}_{0.5}\mathrm{S}{\mathrm{r}}_{0.5}\mathrm{F}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$, and (d) $\mathrm{SrF}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$. (b) The ab-plane dielectric permittivity ${\varepsilon }_{ab}$ for $\mathrm{BaF}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$. The black solid lines are the fitting curves to Eq. (1). The insets are the same plots with a logarithmic scale in temperature.

Figure 3

(a) Schematic illustration of ${\mathrm{Fe}}^{3+}$ off-equatorial displacements in the $\mathrm{Fe}{\mathrm{O}}_5$ bipyramid. The up and down displacements correspond to two $4e$ sites with opposite dipoles and local minima in the energy potential. The calculated energy potentials from Eq. (2) for (b) $U_{\mathrm{Coulomb}}$, (c) $U_{\mathrm{repulsion}}$, and (d) the sum $U_{\mathrm{tot}}$ as a function of off-equatorial ${\mathrm{Fe}}^{3+}$ displacement $z$ in $(\mathrm{Ba},\mathrm{Sr})\mathrm{F}{\mathrm{e}}_{12}{\mathrm{O}}_{19}$. (e) Phase diagram to allow the double-well potential. In the central yellow region, the double-well potential is not allowed. In the left corner (blue), it must appear. In the rest regions (gray), it is possible due to the competition between the Coulomb attraction and Pauli repulsion.

Figure 4

The temperature dependent dielectric permittivity at 1 MHz of (a) the $W$-type hexaferrite $\mathrm{BaC}{\mathrm{o}}_2\mathrm{F}{\mathrm{e}}_{16}{\mathrm{O}}_{27}$ ceramic, (b) the $Z$-type hexaferrite $\mathrm{B}{\mathrm{a}}_{1.5}\mathrm{S}{\mathrm{r}}_{1.5}\mathrm{C}{\mathrm{o}}_2\mathrm{F}{\mathrm{e}}_{24}{\mathrm{O}}_{41}$ single crystal along the $c$ axis, and (c) the $Y$-type hexaferrite $\mathrm{BaSrCoZnF}{\mathrm{e}}_{11}\mathrm{Al}{\mathrm{O}}_{22}$ single crystal along the $c$ axis.