Coupling between Anderson-Type Localized Moments in Relation to the Theory of Transition-Metal Ferromagnetism

When the wave functions of two identical Anderson-type localized moments overlap, the moments tend to align in either parallel or antiparallel configuration. The relative stability of the two configurations is studied in the limit of very small overlap using the Hartree-Fock approximation. It is found that in the highly localized limit, i.e., extremely small $d$-level width, the antiferromagnetic alignment is energetically favored. A necessary condition for ferromagnetic alignment is to have partially occupied $d$ states. Similar consideration can be applied to the $N$-spin problem. If one expands the total energy of the system in terms of the ratio of overlap energy versus intra-atomic Coulomb energy, the lowest-order spin-dependent term has the form of a Heisenberg Hamiltonian. On the other hand, for a ferromagnetically ordered system, the Fermi level is found to fall in the region where the $s$ and $d$ bands hybridize. This indicates that the Fermi surface has strong $d$ character. It will be shown that this model gives a qualitative and roughly quantitative fit to all the important magnetic and electronic properties of iron and nickel.