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

We consider a narrow band constructed from two degenerate orbital states such as dz2 states. Including intra-atomic Coulomb and exchange integrals as well as hopping on a bcc or sc lattice, we find that if n is the number of electrons per atom, for n = 1, the tendency is, for sufficiently narrow bands, for the crystal to break up into two sublattices A and B, with primarily orbital 1 on sublattice A and primarily orbital 2 on sublattice B, and for the spins to line up ferromagnetically. A self-consistent method is used to extend the result beyond the narrow band limit, and the stability of the ferromagnetic state is investigated by calculating the spin wave spectrum as a collective mode. The effect of intra-atomic exchange and orbital degeneracy on the effective exchange parameter is discussed for n≤1.