Calculation of exchange integrals and Curie temperature for La-substituted barium hexaferrites

As the macro behavior of the strength of exchange interaction, state of the art of Curie temperature T_c, which is directly proportional to the exchange integrals, makes sense to the high-frequency and high-reliability microwave devices. Challenge remains as finding a quantitative way to reveal the relationship between the Curie temperature and the exchange integrals for doped barium hexaferrites. Here in this report, for La-substituted barium hexaferrites, the electronic structure has been determined by the density functional theory (DFT) and generalized gradient approximation (GGA). By means of the comparison between the ground and relative state, thirteen exchange integrals have been calculated as a function of the effective value U_eff. Furthermore, based on the Heisenberg model, the molecular field approximation (MFA) and random phase approximation (RPA), which provide an upper and lower bound of the Curie temperature T_c, have been adopted to deduce the Curie temperature T_c. In addition, the Curie temperature T_c derived from the MFA are coincided well with the experimental data. Finally, the strength of superexchange interaction mainly depends on 2b-4f₁, 4f₂-12k, 2a-4f₁, and 4f₁-12k interactions.