Exchange interactions in barium hexaferrite

The electronic structure of $\mathrm{Ba}{\mathrm{Fe}}_{12}{\mathrm{O}}_{19}$ hexaferrite is calculated using the density functional theory and generalized gradient approximation (GGA). The $\mathrm{GGA}+\mathrm{U}$ method is used to improve the description of strongly correlated $3d$ electrons of Fe. The calculation is performed for a number of spin configurations. From differences of the total energies 13 independent exchange integrals are determined as functions of the parameter $U$. Their magnitude decreases with increasing $U$, pointing to the dominating role played by the antiferromagnetic superexchange. The Curie temperature $T_C$ is calculated using the molecular field and the random phase approximations. $T_C$ determined by the random phase approximation agrees with the experimental $T_C$ for $U\approx 6–7\mathrm{eV}$.