Hybrid density functional theory applied to magnetite: Crystal structure, charge order, and phonons

Abstract

The electronic structure and equilibrium structure of magnetite $({\text{Fe}}_{3}{\text{O}}_{4})$ in the high temperature cubic $Fd\overline{3}m$ and low temperature monoclinic $P2/c$ unit cells have been computed using the Perdew-Wang generalized gradient approximation (GGA) to density functional theory (DFT) and the B3LYP hybrid density functional. The ground state for the GGA-DFT is an itinerant electron metallic state in the cubic unit cell and the ground state for the B3LYP functional is a charge ordered semiconducting state in the monoclinic unit cell. The equilibrium structure predicted by the B3LYP functional for ${\text{Fe}}_{3}{\text{O}}_{4}$ in the $P2/c$ unit cell has been calculated with lattice parameters fixed at values obtained in recent x-ray diffraction work and with the lattice fully relaxed. Bond lengths obtained with lattice parameters fixed at experimental values are in excellent agreement with x-ray measurements [J. P. Wright et al., Phys. Rev. B 66, 214422 (2002)]. The degree of charge order, measured as disproportionation of charge on octahedral B sites, is considerably less than unity and in reasonable agreement with values from resonant x-ray diffraction measurements. However, conduction electrons are found to be fully localized on B1 and B4 sites in orbitally ordered ${t}_{2g}$ states. This shows that they are formally ${\text{Fe}}^{2+}$ ions while Fe B2 and B3 sites are formally ${\text{Fe}}^{3+}$ sites. Therefore Verwey's original conjecture regarding charge localization in ${\text{Fe}}_{3}{\text{O}}_{4}$ applies, even though the specific pattern of charge order is different. GGA-DFT and B3LYP density functionals were used to calculate phonons at the $\ensuremath{\Gamma}$ point of the Brillouin zone. Phonon frequencies predicted for these crystal structures are compared to frequencies from infrared conductivity and Raman scattering experiments. Charge ordering causes symmetry breaking of force constants on symmetry lowering from the cubic $Fd\overline{3}m$ unit cell to the $P2/c$ unit cell. This produces frequency splitting of modes which are degenerate in the cubic unit cell and concentration of ion displacements in phonon eigenvectors on particular Fe octahedral B site chains, especially in the highest frequency bands.