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The aim of this work is to unravel a basic microscopic picture behind complexmagnetic properties of hexagonal manganites. For these purposes, we considertwo characteristic compounds: YMnO3 and LuMnO3, which form different magneticstructures in the ground state. First, we establish an electronic low-energymodel, which describes the behavior of the Mn 3d bands of YMnO3 and LuMnO3, andderive parameters of this model from the first-principles calculations. Fromthe solution of this model, we conclude that, despite strong frustrationeffects in the hexagonal lattice, the relativistic spin-orbit interactions liftthe degeneracy of the magnetic ground state so that the experimentally observedmagnetic structures are successfully reproduced by the low-energy model. Then,we analyze this result in terms of interatomic magnetic interactions, whichwere computed using different approximations (starting from the modelHamiltonian as well as directly from the first-principles electronic structurecalculations in the local-spin-density approximation). We argue that the mainreason why YMnO3 and LuMnO3 tend to form different magnetic structures isrelated to the behavior of the single-ion anisotropy, which reflects thedirectional dependence of the lattice distortion: namely, the expansion andcontraction of the Mn-trimers, which take place in YMnO3 and LuMnO3,respectively. On the other hand, the magnetic coupling between the haxagonalplanes is controlled by the next-nearest-neighbor interactions, which are lesssensitive to the direction of the trimerization. Finally, using the Berry-phaseformalism, we evaluate the magnetic-state dependence of the ferroelectricpolarization, and discuss potential applications of the latter inmagnetoelectric switching phenomena.

Magnetic structure of hexagonal YMnO3 and LuMnO3 from a microscopic point of view