Magnetic Anisotropy of Europium in Iron Garnet at Low Temperatures

Abstract

Trivalent europium (${\mathrm{Eu}}^{3+}$) has an anomalous magnetic behavior at low temperatures because its first magnetic state is removed by approximately 500°K from the nonmagnetic ground state. The effect of a crystal field or a europium-iron (Eu-Fe) exchange interaction on the energy of the ground state is the result of the partial breakdown of the $\mathrm{LS}$ coupling caused by these perturbations. We calculate the first-order cubic anisotropy constant $K_1$ for the ground state of ${\mathrm{Eu}}^{3+}$ in europium iron garnet (EuIG) due to the combined action of the Eu-Fe exchange and crystalline field. The use of the single-ion (molecular-field) approximation permits us to replace the Eu-Fe exchange interaction by an effective exchange field acting on the ${\mathrm{Eu}}^{3+}$ ions; the assumption is made that the anisotropy in the Eu-Fe exchange does not make a major contribution to the anisotropy of the ground state. We find that the isotropic exchange field and fourth-rank crystalline field first contribute to the cubic anisotropy of the ground state of ${\mathrm{Eu}}^{3+}$ in the fifth order of the energy; likewise the second-rank crystalline field contributes in the sixth order. The fifth- and sixth-order perturbation formulas for a Hermitian operator have been derived and are given in an Appendix. We calculate the anisotropy energy for a single ${\mathrm{Eu}}^{3+}$ ion in EuIG and average the results over the six magnetically inequivalent sites of the garnet structure. The order of magnitude of the calculated anisotropy constant $K_1$ is in agreement with the experimental data that are available at low temperatures, but detailed comparison is not possible at present because of a relatively large number of poorly known parameters.