Magnetic charge and magnetoelectricity in hexagonal manganites and ferrites

Magnetoelectric (ME) materials are of fundamental interest and are investigated for their broad potential for technological applications. Commonly the dominant contribution to the ME response is the lattice-mediated one, which is proportional to both the Born electric charge $Z^{\rm e}$ and its analogue, the dynamical magnetic charge $Z^{\rm m}$.\footnote{J. \'{I}\~{n}iguez, Phys. Rev. Lett. {\bf 101},117201 (2008).} A previous study has shown that exchange striction acting on noncollinear spins induces much larger magnetic charges than when $Z^{\rm m}$ is driven by spin-orbit coupling.\footnote{M. Ye and D. Vanderbilt, Phys. Rev. B {\bf 89}, 064301 (2014).} The hexagonal manganites $R$MnO$_3$ and ferrites $R$FeO$_3$ (R$=$Sc, Y, In, Ho-Lu) exhibit strong couplings between electric, magnetic and structural degrees of freedom, with the transition-metal ions in the basal plane are antiferromagnetically coupled through super-exchange so as to form a 120$^\circ$ spin arrangement. Here we present a theoretical study of the magnetic charges, and of the spin-lattice and spin-electronic ME constants, in these hexagonal manganites and ferrites, clarifying the conditions under which exchange striction leads to enhanced $Z^{\rm m}$s and anomalously large in-plane spin-lattice ME effects.