Simple Narrow-Band Model of Ferromagnetism Due to Intra-Atomic Exchange

As a model of ferromagnetism due to intra-atomic exchange, we consider a narrow band with two orbital states, such as $d_{z2\mathrm{}}$ states in a bcc lattice. We include hopping and intra-atomic Coulomb and exchange terms in the Hamiltonian. If we assume that the hopping process does not change the orbital state, it is found for one electron per atom and for sufficiently narrow bands that it is energetically favorable for two sublattices to form, each with predominantly one of the orbital states, and for the spins to line up ferromagnetically. We use the random-phase approximation to extend this result beyond the narrow-band limit. The stability of this ferromagnetic state is investigated by calculating its spin-wave spectrum, and the region of stability as well as the effective exchange parameter depend on the strength of the intra-atomic exchange integral. For one electron per atom, the mechanism is too weak for real ferromagnets, but it becomes stronger for less than one electron per atom.