Electronic structure and conductivity of ferroelectric hexaferrite:Ab initiocalculations

${\mathrm{Ba}}_{0.5}{\mathrm{Sr}}_{1.5}{\mathrm{Zn}}_2{\mathrm{Fe}}_{12}{\mathrm{O}}_{22}$ is a promising multiferroic compound in which the electric polarization is intimately connected to the magnetic state. In principle, ferroelectrity might exist above the room temperature, but the electrical conductivity that increases with increasing temperature limits it to temperatures below $\approx 130\mathrm{K}$. We present results of an ab initio electronic structure calculation of the $\left(\mathrm{BaSr}\right){\mathrm{Zn}}_2{\mathrm{Fe}}_{12}{\mathrm{O}}_{22}$ system. To improve the description of strongly correlated $3d$ electrons of iron, the $\mathrm{GGA}+U$ method is used. The results show that the electrical conductivity strongly depends on relative fractions of iron and zinc in the tetrahedral sublattice that belongs to the spinel block of the hexaferrite structure. If this sublattice is fully occupied by zinc, the system is an insulator with a gap of $\approx 1.5\mathrm{eV}$. If it is occupied equally by Fe and Zn the gap decreases by a factor of 2, and the system is metallic when this sublattice is filled by iron only.