Exchange-Perturbation Treatment of Magnetic Ordering in Nonconducting Solids

Abstract

Many nonconducting ionic solids with open-shell $3d$ or $4f$ cations exhibit magnetic order at low temperatures. In this paper such ordering is analyzed on the basis of exchange-perturbation theory applied to a complex of two paramagnetic cations and one diamagnetic anion in different geometric configurations. The perturbation energy is then summed over all complexes in the solid, for a given structure and for different spin patterns. The unperturbed Hamiltonian is chosen such that it is invariant under permutations of all electrons of the complex; the unperturbed ground-state wave function is an eigenfunction of this Hamiltonian. Full permutation symmetry is thus taken into account from the outset. On the other hand, space-group (or point-group) symmetry is not considered. An effective-electron model with four electrons on three centers is used to evaluate the influence of a diamagnetic anion on the interactions between unpaired electrons of two cations (superexchange). In addition, the effect of the valence shells of the cations on the cation-cation interaction is taken into account on the basis of a two-center six-electron model. The perturbation energies are evaluated in first and second orders, using Gaussian wave functions for the orbitals of the effective electrons. It is found that the model predicts both ferromagnetic and antiferromagnetic orderings, the type of ordering depending upon the extensions of the orbitals for the different effective electrons, the lattice parameters, and the symmetry of the crystal. It is established that exchange between unpaired electrons via the anion, when summed over the crystal, generally favors antiferromagnetic alignment of the spins, whereas indirect exchange via the cation valence shells leads to ferromagnetism. The model is applied to magnetic ordering in the Mn and Eu chalcogenides. Specifically, EuO and EuS are found to be ferromagnetic, whereas the remaining Eu salts are antiferromagnetic of the second kind. For the Mn compounds the indirect exchange via cation valence shells does not affect the stable magnetic structure, which is then determined by (antiferromagnetic) superexchange. Néel and (Curie temperatures for the Mn and Eu chalcogenides calculated on the basis of this model are (in fair agreement with experiment. A detailed calculation of the coupling parameters $J_1$ and $J_2$ for the systems MgO: ${\mathrm{V}}^{2+}$ and MgO: ${\mathrm{Mn}}^{2+}$ is made and the results are compared both with experiments and with recent configuration- interaction analyses. The observed positive (ferromagnetic) sign of $J_1$ in MgO: ${\mathrm{V}}^{2+}$ is reproduced by the model.