Electronic structure and exchange interactions in the manganese-based pyrochlore oxides

We describe the ferromagnetism in the manganese pyrochlores, such as ${\mathrm{Tl}}_{2}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7}$ and ${\mathrm{Sc}}_{2}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7},$ in terms of the interplay between superexchange, Zener double exchange, and indirect exchange, the first being antiferromagnetic (AF) while the last two are ferromagnetic. The tendency towards the antiferroalignment of the localized Mn spins is significantly weakened due to (i) the presence of spin frustration in the Mn sublattice and (ii) the large bend in the Mn-O-Mn bond, which reduces the magnitude of the AF superexchange. This helps the ferromagnetic interaction terms, viz., the indirect exchange and the Zener double exchange, to dominate. The indirect exchange on the Mn-O-Mn bond, which is modeled by a coupling of the O $2p$ electrons to the conduction-band states, produces a weak ferromagnetism in the insulating pyrochlores. In the metallic pyrochlores, the indirect exchange is also present, but it is now supplemented by the Zener double exchange, making the ferromagnetism more robust, as indicated from a higher value of the Curie temperature ${T}_{c}.$ Density-functional calculations for the metallic ${\mathrm{Tl}}_{2}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7}$ pyrochlore, show the itinerant Zener carriers mediating double-exchange to be electrons in the ${\ensuremath{\Gamma}}_{1}$ minority spin band crossing the Fermi energy, with predominantly Mn $3d,$ Tl $6s,$ and O $2p$ characters. These Zener carriers move in a lattice of localized Mn ${t}_{2g}$ spins, with the carrier spins aligned antiparallel to the localized spins.