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Improper ferroelectricity (trimerization) in the hexagonal manganitesRMnO$_3$ leads to a network of coupled structural and magnetic vortices thatinduce domain wall magnetoelectricity and magnetization neither of which,however, occurs in the bulk. Here we combined first-principles calculations,group-theoretic techniques, and microscopic spin models to show how thetrimerization not only induces a polarization but also a bulk magnetization andbulk magnetoelectric (ME) effect. This results in the existence of a bulklinear ME vortex structure or a bulk ME coupling such that if direction ofpolarization reverses so does magnetization. To measure the predicted MEvortex, we suggest RMnO$_3$ under large magnetic field. We suggest a family ofmaterials, the hexagonal RFeO$_3$ ferrites, also display the predictedphenomena in their ground state.

Bulk magnetoelectricity in the hexagonal manganites and ferrites