Anisotropic Superexchange inβMnS

The suggestion by Keffer of an anisotropic superexchange of the Moriya type in $\beta \mathrm{MnS}$ led to the present detailed calculation. Whereas Moriya's work was confined to ions with one electron in the $3d$ shell and to cases of low symmetry, this calculation was performed for a half-filled $3d$ shell and tetrahedral symmetry. The unperturbed state was taken to consist of two ${\mathrm{Mn}}^{++}$ ions together with their common ${\mathrm{S}}^{--}$ nearest neighbor, each in a tetrahedral crystal field. The perturbing Hamiltonian contained the electronic spin-orbit interaction and the Coulomb interaction between electrons on neighboring ions. The mechanism involved going from the ground state to an excited crystal field state on one ${\mathrm{Mn}}^{++}$ ion by the spin-orbit effect, then back to the ground state by superexchange between the Mn's. The direction of the D vector was found to be perpendicular to the plane of the three ions, and its magnitude of the order of a superexchange integral times the spin-orbit parameter over a crystal field splitting energy. The existence of the effect depends importantly upon $\pi$ overlap between the Mn and S orbitals.