Analysis of the Difference between the Pyroxenes LiFeSi2O6 and LiFeGe2O6 in Their Spin Order, Spin Orientation, and Ferrotoroidal Order

The pyroxenes LiFeSi2O6 and LiFeGe2O6 are isostructural and isoelectronic, but they differ in their spin order, spin orientation and ferrotoroidal order. The reasons for these differences were probed by density functional theory electronic structure calculations and magnetic dipole–dipole (MDD) interaction energy calculations. The ferromagnetic Fe-zigzag chains of LiFeSi2O6 arise from the antiferromagnetic interchain spin exchanges and the antiferromagnetic Fe-zigzag chains of LiFeGe2O6 from the antiferromagnetic intrachain spin exchange. The preferred spin orientations of LiFeGe2O6 and LiFeSi2O6 are not caused by spin–orbit coupling but by MDD interactions. LiFeSi2O6 undergoes a ferrotoroidal order because it has spin exchange rings made up of an even number of antiferromagnetic spin exchanges paths with comparable strengths. A ferrotoroidal order is not expected for LiFeGe2O6 because it has no such spin exchange rings.